<p>In this lesson, students will practice using existing formulas to calculate the perimeter and area of polygons scaffolding from Lesson 1 on circumference and area of circles. Students will compute the perimeter and area of compound figures made up of three and four-sided polygons. Students will:&nbsp;<br>- understand the relationship between perimeter and circumference.&nbsp;<br>- find the perimeters of triangles and quadrilaterals.&nbsp;<br>- calculate the area of triangles and quadrilaterals (using a formula reference page).&nbsp;<br>- describe strategies for determining the perimeter and area of compound figures.&nbsp;<br>- identify which dimensions are necessary for various perimeter and area calculations in polygons and compound figures.&nbsp;<br>- use perimeter and area concepts to calculate the perimeter and area of compound figures.&nbsp;<br>- use a coordinate graph to represent different triangles and quadrilaterals.&nbsp;<br>- use coordinate grids to calculate perimeter and area.&nbsp;</p>

<p>- How are relationships represented mathematically?<br>- How may data be arranged and represented to reveal the relationship between quantities?&nbsp;<br>- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?<br>- How can mathematics help us communicate more effectively?<br>- How can probability and data analysis be used to make predictions?<br>- How does the type of data effect the display method?<br>- How can mathematics help to quantify, compare, depict, and model numbers?<br>&nbsp;</p>

<p>- Parallelogram: A quadrilateral with opposites sides parallel and congruent.&nbsp;<br>- Polygon: A geometric figure created with three or more straight segments that meet at their endpoints.&nbsp;<br>- Quadrilateral: A polygon with four sides.&nbsp;<br>- Rectangle: A quadrilateral with opposites sides parallel and four right angles.&nbsp;<br>- Square: A quadrilateral with four congruent sides, opposite sides parallel, and four right angles.&nbsp;<br>- Trapezoid: A quadrilateral with one pair of parallel sides.&nbsp;<br>- Triangle: A polygon with three sides.&nbsp;<br>- Radius: A segment from the center of a circle to the edge.</p>

<p>- student copies of Vocabulary Journal pages (M-7-6_Vocabulary Journal)&nbsp;<br>- student copies of the Lesson 2 Entrance Ticket (M-7-6-2_Lesson 2 Entrance Ticket and KEY)&nbsp;<br>- student copies of Areas Review (M-7-6-2_Areas Review and KEY)&nbsp;<br>- student copies of Perplexing Polygons (M-7-6-2_Perplexing Polygons and KEY)&nbsp;<br>- student copies of the Partner Quiz (M-7-6-2_Partner Quiz and KEY)&nbsp;<br>- Compound Figures (M-7-6-2_Compund Figures and KEY)&nbsp;<br>- Formula Sheet for Areas of Polygons (M-7-6-2_Formula Sheet for Areas of Polygons)&nbsp;<br>- Optional Refresher–Graphing on Coordinate Grid (M-7-6-2_Optional Refresher-Graphing on Coordinate Grid)&nbsp;<br>- construction paper or posterboard triangle and quadrilateral cutouts of a variety of shapes and sizes (length and width of 3–10 cm) for students to trace&nbsp;<br>- small paper or plastic dots, about 10–15 per student stored in zipper bags (small colored paper dots from a hole punch work well)&nbsp;<br>- chart paper and markers for group use&nbsp;<br>- centimeter grid paper for student use&nbsp;<br>- centimeter rulers&nbsp;<br>- transparency of centimeter grid paper&nbsp;<br>- student copies of the Small Group activity page (M-7-6-2_Small Group Record)&nbsp;<br>- student copies of Fabulous Formations (M-7-6-2_Fabulous Formations and KEY)&nbsp;<br>- student copies of Create a Character record sheet (M-7-6-2_Create a Character)&nbsp;<br>- blank coordinate planes (M-7-6-2_Blank Coordinate Plane)</p>

<p>- Observe students throughout the Reviewing Areas of Polygons and Investigating Compound Figures activities to informally evaluate their understanding.&nbsp;<br>- Evaluate students' comprehension using the Perplexing Polygons activity and presentations.&nbsp;<br>- At the end of the class, use the Partner Quiz to assess students' understanding.&nbsp;<br>&nbsp;</p>

<p>W: Students will first complete the Lesson 2 Entrance Ticket. Next, you ask students to describe a location, such as a student's location in the classroom or the school's location in the city. Students are given a piece of centimeter meter grid paper to begin the lesson. To begin discussion about coordinate axes, you ask them to draw both horizontal and vertical number lines on the paper, how the x-axis describes horizontal positions and the y-axis describes vertical positions.<br>H: Demonstrates the ability to locate or create points on the coordinate plane using verbal instructions such as left, right, up, and down, and moving to using coordinate pairs (x, y). The exploration of connecting various points to form figures such as a triangle, rectangle, or parallelogram leads to the concepts of perimeter and area of triangles and quadrilaterals as the lesson progresses.&nbsp;<br>E: Students draw their own axes on centimeter grid paper to plot points that you suggest. Understanding the placement of the x- and y-coordinates is important. Several points are plotted as a class. Students are given numerous more sets of ordered pairs to graph, which will form polygons when connected. The class investigates the perimeter and area of various figures using grids. This leads to the use of formulas for triangles and quadrilaterals. The amount of time and detail required for this is determined by the student's responses on the Entrance Ticket. The class completes the Areas Review sheet, which serves as a formula page for the remainder of the unit. Pairs of small groups complete the Investigating Compound Figures activity. By the end of the lesson, students problem solve in situations requiring many calculations for compound figures including triangles and quadrilaterals in the Perplexing Polygons activity and present their strategies and solutions.&nbsp;<br>R: Students are encouraged to change their solutions during work and class discussions. You and your student peers ask questions to help partners and presenters identify inaccuracies in their thinking and correct them. Students are encouraged to write down additional ideas to their solutions as other groups present them in order to improve their own problem solving and develop a better understanding of how many ways might work.<br>E: Throughout class discussions and student work time, you can conduct informal assessments of student understanding through observation. When students submit one or more Perplexing Polygons tasks, their comprehension of the perimeter and area of polygons and compound figures is examined. Each pair of students takes a partner quiz to assess their level of understanding and to help select extension activities for remediation or enrichment.&nbsp;<br>T: The lesson can be adjusted to match the needs of the students by using the extension recommendations. The small group activity is appropriate for students who require further assistance, while the expansion can be used for students who demonstrate proficiency. Additional activities are proposed for classroom stations and the use of technology.<br>O: Students learn vocabulary terms for coordinate graphing, triangles, and quadrilaterals. Students learn graphing coordinate pairs in all four quadrants, as well as simple figures like polygons, which will be used to scaffold in Lesson 3. Students use the grids on the coordinate graph to calculate the perimeter and area of basic geometric shapes before moving on to formulas for triangles and quadrilaterals. Students work together to solve problems related to calculating the perimeter and area of compound figures and then present their findings to the class. The lesson ends with a partner quiz and optional extension activities. This lesson prepares students to move into the concepts of congruence, similarity, and scale factor in Lesson 3.</p>

<p>As students enter the room, distribute a Lesson 2 Entrance Ticket (M-7-6-2_Lesson 2 Entrance Ticket and KEY). Make sure students may obtain a PSSA or other formula sheet. Ask students to spend 4-8 minutes working individually on their Entrance Ticket. Assure students that you are simply trying to find out what they already know and that they may not know how to do everything. Use the student responses on the entrance ticket to determine the pace and depth at which you teach or review perimeter and area concepts later in the lesson. It may be helpful to have students score the perimeter and area calculations so that you may make a quick review by scanning through the Entrance Tickets while students have small spans of work time during the early part of the lesson.<br><br>An optional refresher on how to graph using the coordinate plane is available (M-7-6-2_Optional Refresher-Graphing on Coordinate Grid).&nbsp;<br><br><strong>Introducing Perimeter and Area of Graphed Figures Activity&nbsp;</strong><br><br><strong>"Let's graph some more points together."</strong> Hand out the coordinate plane worksheet (M-7-6-2_Blank Coordinate Plane). Say, <strong>"First, graph (1, 1), (6, 1), (6, 4), and (1, 4)." "Connect the points in this order as you graph them."</strong>&nbsp;<br><br>Allow students time to graph. Then ask:<br><br><strong>“What figure was created?”</strong> (<i>rectangle</i>)&nbsp;<br><strong>"How many units is it around the outside of our rectangle?"</strong> (<i>16 cm</i>)<br><strong>"You discovered the perimeter of our rectangle. Because perimeter is a measurement of distance, your label should be in centimeter. How did you determine your answers?"</strong> (<i>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>)&nbsp;<br><strong>"I also want you to calculate the area, or space inside the rectangle."</strong> (<i>15 square centimeters</i>)&nbsp;<br><strong>"How did you find your answer?"</strong> (<i>counted the grid squares inside, multiplied length by width, or multiplied the number of rows by the number of columns</i>)<br><strong>"Some of you used grids to help you, while others used a different strategy, such as a formula. Let's practice some additional examples. After graphing these, you can use any approach to calculate the areas and perimeters. Use the grids on your paper to help you. We'll work with the formulas for these figures in the following part of our lesson."&nbsp;</strong><br><br>Give students 3-5 more sets of ordered pairs to make triangles and quadrilaterals. Allow them 5-7 minutes to graph them and calculate the perimeters and area. Make sure to teach students strategies for working with the angled sides. For example, given a parallelogram, the area can be determined 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 represented as the larger rectangle or parallelogram that contains them, which is then divided in half to determine area. Allow students to check their answers with a partner after 5 minutes. Check for accuracy by walking around to monitor and inviting students to come to the board or overhead to display their responses at the end of the work time.<br><br><strong>Reviewing Areas of Polygons Activity</strong><br><br>At this point in the lesson, decide how much time should be spent studying or reviewing the area formulas of triangles and quadrilaterals. Even if students already have a good comprehension, spend a few minutes reviewing it. If students struggled with the Entrance Ticket questions, spend more time explaining the formulas and why they work.&nbsp;<br><br><strong>"Using your Entrance Tickets, I was able to observe that this class is already proficient in finding</strong> (perimeters of ____ and areas of ____). <strong>What we need to </strong>(work on/ or review) <strong>is how to find </strong>(perimeters of _____ and areas of _____)<strong>."</strong><br><br><strong>"In any polygon or figure with straight sides, we simply go around its outside and sum the lengths of each side. What if a figure contained another marking, such as an altitude or height marking inside? Do we include that in the perimeter, too?"</strong> (For example, draw a triangle with 3 side measures and an altitude line with a measure inside.) (<i>No, only 3 sides are provided.</i>)&nbsp;<br><br><strong>"How about for the area? What measurements are required?"</strong> (<i>It varies depending on the figure, although it usually only includes the base and perpendicular height, not the angled sides.</i>)<br><br>Distribute the Areas Review sheet (M-7-6-2_Areas Review and KEY). This sheet will be completed and used as a formula page for the rest of the unit. The Formula Sheet for Areas of Polygons (M-7-6-2_Formula Sheet for Areas of Polygons) is another option that can be used later in class.&nbsp;<br><br>Either work through the examples as a class, assign students in pairs to work on one polygon and present it, or offer students time with a partner to finish it. Ensure that all students have the opportunity to check and correct their responses. If students' Entrance Tickets indicate that they are having difficulties understanding the figures, names of their parts, or computing areas using the formulas, spend extra time here teaching these concepts. It is expected that many students will review this material. This would also be an ideal time to review the questions on the Entrance ticket with the class, or have students who correctly solved the questions present their solutions. A solid understanding of these figures and formulas is required to progress to the next step in the lesson and Lesson 3.<br><br><strong>Investigating Compound Figures Activity</strong><br><br>Tell students, <strong>"In our last activity, we worked on calculating the perimeter and area of triangles and quadrilaterals. In this activity, I will challenge you by combining some of the figures you already know into compound figures you may be unfamiliar with. You'll need to apply your problem-solving abilities to complete the calculations. Try to determine the perimeter and area of this figure."</strong> Give students 1-3 minutes to provide methods and solutions.</p><figure class="image"><img style="aspect-ratio:296/175;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_71.png" width="296" height="175"></figure><p>There will probably be a range of strategies. Make sure that these strategies are discussed by a student or yourself:&nbsp;<br><br><strong>Perimeter option 1:</strong> Subtract the vertical measurements to get the missing vertical length (12 – 3 = 9), and subtract the horizontal lengths to get the missing horizontal value (15 – 9 = 6). Then add all sides.</p><figure class="image"><img style="aspect-ratio:212/163;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_72.png" width="212" height="163"></figure><p><strong>Perimeter option 2:</strong> Note that all vertical segments on the right side of the figure can be slid together to form a segment the same length as the left vertical side (12), so vertical measures will be 12 + 12 as in a rectangle. 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 calculated as 2(12 + 15) or 2(12) + 2(15).<br><br><strong>Area option 1:</strong> To find the area, divide the figure into two rectangles with a line inside. A vertical line can be used to get a 9 x 12 and a 3 x 6 rectangle, while a horizontal line can be used to get a 3 x 15 and a 9 x 9 rectangle. In any scenario, calculate the area of each rectangle and add the two values.</p><figure class="image"><img style="aspect-ratio:488/178;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_73.png" width="488" height="178"></figure><p><strong>Area option 2:</strong> Consider this a large rectangle with a piece missing (or cut out of) one corner. Outline the larger rectangle. Calculate both the large outlined rectangle and the rectangular piece that is "cut out". To remove the smaller cut-out from the area computation, subtract it from the larger rectangle.</p><figure class="image"><img style="aspect-ratio:224/165;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_74.png" width="224" height="165"></figure><p>Ask students to practice another example. This one is more difficult.</p><figure class="image"><img style="aspect-ratio:176/143;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_75.png" width="176" height="143"></figure><p>Answers:&nbsp;<br><br>P = (5)(6) = 30 cm<br><br>A = (3.5)(9) ÷ 2 + (9 + 6) ÷ 2 • (5.5) = 57 cm²<br><br>There may be several correct strategies suggested. Make sure you discuss dividing the pentagon into a trapezoid and a triangle. This strategy involves calculating the area of the trapezoid and triangle separately and then combining them. The perimeter is calculated by multiplying 5 times 6 centimeters because the tic marks show that all sides are congruent.<br><br>Continue to provide enough examples that students feel confident in developing strategies to work with compound figures. Students will be working with a partner to practice additional problems like these in the Perplexing Polygons activity, so the number of additional examples may be limited.<br><br>The figures below show other shapes that you can use before or after the trapezoid example. Insert measurements to meet your students' needs. Use whole numbers for students who are struggling with the calculations or strategies. Choose decimal or fractional values for students who need a challenge.</p><figure class="image"><img style="aspect-ratio:533/121;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_76.png" width="533" height="121"></figure><p>(<strong>Note:</strong> This example is two polygons.)<br><br>For optional group or individual practice with compound figures, use the Compound Figures worksheet (M-7-6-2_Compund Figures and KEY.docx).<br><br><strong>Partner Practice: Perplexing Polygons Activity</strong><br><br>If students have previously worked independently, group them into pairs. Give each student the Perplexing Polygons sheet (M-7-6-2_Perplexing Polygons and KEY). Encourage partners to describe how they will solve and label each problem and why. They may use the strategies already taught in class as examples, or they can think of their own strategies. Remind the students that each pair will be chosen at random to present one problem to the class. Allow about 15-20 minutes of work time before presentations. Provide support as needed during work time. As the work time draws to a conclusion, remind students that their work should be displayed and answers should be correctly labeled.&nbsp;<br><br>At the end of the work time, use a strategy like drawing numbers or rolling a number to assign a problem to each pair to present. During presentations, invite audience members to provide comments to help presenters who are having trouble expressing their work or have made mistakes. Allow students to make changes or additions to their solutions after the presentations.<br>After the presentations, review the perimeter and area strategies covered in this lesson for compound figures and correct any remaining misconceptions. Respond to any student questions that arise.&nbsp;<br><br><br><strong>Partner Quiz</strong><br><br>With the students still in pairs, administer the Partner Quiz (M-7-6-2_Partner Quiz and KEY). Allow students about 8-15 minutes to finish the quiz. Use the results to determine the best extension activities for each student.&nbsp;<br><br><strong>Extension:</strong><br><br>Discuss how important it is to comprehend and use the appropriate vocabulary words while communicating mathematical ideas. During this lesson, students should record the following terms in their vocabulary journals: <i>coordinate graph, kite, ordered pair, parallelogram, quadrant, quadrilateral, rectangle, rhombus, square, trapezoid, triangle, x axis, y axis</i>. Keep a supply of Vocabulary Journal pages on hand so that students can add them as needed. Bring up perimeter and area examples from throughout the school year. Use polygon examples to teach ratio and proportional reasoning, as well as similarity and scale factor units. Ask students to bring in pictures or items that are compound figures and require the strategies from this unit. Display these examples for students to work on. Discuss the use and meaning of such instances in each specific circumstance. Continue to distinguish between labeling perimeters in standard units and areas in square units. Require students to use appropriate labeling in both verbal and written responses.<br><br><strong>Small Group: Revisiting the Compound Polygons:</strong> Use this activity for the entire class, or for students who struggled with compound figure perimeters and areas throughout the lesson or on the Partner Quiz. Provide students with centimeter grid paper. Make a variety of triangles and quadrilateral shapes ranging from 3 to 10 centimeters in length and width. Cut out the figures from poster board material or construction paper.&nbsp;<br>Ask students to trace two of the paper cut-outs, lining up the edges without gaps or overlaps to create a compound shape as they trace. Students will benefit from lining the figures along grid lines on their paper.&nbsp;<br>Students should repeat this with other shapes until they have traced three or four compound figures.<br>Teach students how 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 help them determine the lengths of the sides, or they can use a centimeter ruler.&nbsp;<br>Encourage students to apply the area formulas and strategies from the lesson to calculate the area of their figures. After calculating the areas, students should be encouraged to double-check their work by counting the number of unit squares represented inside their compound figures on the grid paper.&nbsp;</p><p><i>Optional:</i> Have students draw a little drawing of the compound figure, its dimensions, and their perimeter and area calculations on the Small Group Record sheet (M-7-6-2_Small Group Record) and turn it in after the activity.<br>Remind students to indicate their perimeter answers with centimeter label and their areas in square centimeter labels. If time allows, instruct students to carefully trace a new compound figure using three of the paper cut-outs, with one figure placed inside the others to remove some of the area. Instruct them to determine the perimeter and area of this more complex figure.&nbsp;</p><p><strong>Fabulous Formations:</strong> Students who shown proficiency in computing the perimeters and areas of triangles, quadrilaterals, and compound figures can solve problems involving compound figures with circles, semi-circles, and/or more than two polygons.&nbsp;<br>Instruct students to use the 3.14 approximation for pi and round their answers to the nearest hundredth. Calculators can be used for this activity.<br><br>Give each student a copy of Fabulous Formations (M-7-6-2_Fabulous Formations and KEY). Allow students to collaborate with a partner or in small groups.&nbsp;<br><br>If time permits, have students working on this extension activity present one problem to the class. Instruct them to assist the class in solving it by asking for input from the class and taking them through any parts that the class is unsure about.&nbsp;<br><br><strong>Station: Create a Character:</strong> Give students centimeter or \(1 \over 4\) inch grid paper and markers. Create a Character record sheet (M-7-6-2_Create a Character).&nbsp;<br>Instruct students to draw and mark the x- and y-axis in the center of the page. On their grid paper, have students create a cartoon-style character using polygons and compound figures. They should take care to use the grid paper's intersection points as vertex points for the polygons.&nbsp;<br><br>Students then identify a starting point and generate a list of coordinate pairs required to create their character on their record sheet. The ordered pairs must be listed in the order in which they are plotted, and when connected in order, they will form the characters' outline. When a new section or detail of the figure should not be connected to the previous point, students should use phrases like "stop here" or "lift your pencil here." If time allows, students can exchange lists of coordinate pairs to draw another student's character on a new piece of grid paper.<br><br><strong>Technology: Locate It!</strong> If computers are available for student usage, students can practice coordinate graphing by completing activities on mathematics education Web sites. Instruct students to visit one of the following sites, or one you choose:&nbsp;<br>A game in which students name coordinates to locate aliens.&nbsp;<br><a href="http://www.mathplayground.com/locate_aliens.html">http://www.mathplayground.com/locate_aliens.html</a>&nbsp;<br><br>A game in which students choose a coordinate pair from a list of four options to catch a mole, with several difficulty levels.&nbsp;<br><a href="http://funbasedlearning.com/algebra/graphing/points3/"><span style="color:#1155cc;"><u>http://funbasedlearning.com/algebra/graphing/points3/</u></span></a>&nbsp;</p>