# Mathematics as Reasoning Students will use reasoning skills to determine the best method for maximizing area.

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1 Title: A Pen for Penny Brief Overview: This unit is a reinforcement of the concepts of area and perimeter of rectangles. Methods for maximizing area while perimeter remains the same are also included. The storyline gives a real-world problem while allowing students to use mathematics to perform the given task. Conclusively, the students will determine the best method for maximizing area, make a written conclusion regarding maximizing area, and be assessed. Links to NCTM Standards: Mathematics as Problem Solving Students will use problem-solving strategies to investigate the effect perimeter has on area. Students will apply formulas for the perimeter and area of a rectangle to real-world situations. Mathematics as Communication Students will communicate their ideas verbally within their groups while using manipulatives and performance tasks, as well as in written form throughout the unit. Mathematics as Reasoning Students will use reasoning skills to determine the best method for maximizing area. Mathematical Connections Students will make connections between geometry and basic algebraic equations while solving real-life situations. Links to Maryland High School Mathematics Core Learning Goals: The student will recognize, describe, and extend patterns and functional relationships that are expressed numerically, algebraically, and geometrically The student will solve problems using two-dimensional figures The student will use techniques of measurement and will estimate, calculate, and compare perimeter and area of two- and three-dimensional figures and their parts. The results will be expressed with appropriate precision. Grade/Level: This unit is appropriate for students in Grades 8-12 enrolled in Pre-Algebra, Algebra I, and Geometry. Duration/Length: Three to four days (variable)

2 Prerequisite Knowledge: Objectives: Students should have working knowledge of the following skills : Calculating the area and the perimeter of a rectangle Evaluating algebraic expressions Solving algebraic equations Students will be able to: determine the best shape that will maximize area and minimize perimeter. apply algebraic/geometric expressions to solve real-world problems. Materials/Resources/Printed Materials: Pencil TI-83 Graphing Calculator Straightedge Warm-up and activity sheet Development/Procedures: During this unit, students will work with group members or partners to determine whether or not area can change and perimeter remain the same, while strengthening their understanding of area and perimeter. In conclusion they will determine the shape which will always maximize area. Day One: Students will complete the warm-up activity (Day 1 warm-up) to reinforce the idea of area being multiplication and perimeter being addition. A group manipulative activity (Activity 1) will also be completed. Day Two: Students will complete a warm-up (Day 2 warm-up) to recall yesterday s lesson. At this point the teacher will introduce and give instructions for problem #1. Day Three: Students will need to complete problem #2 and #3. (Teacher must give instruction for graphing calculator use for problem # s 2 and 3.) Warm-ups and templates for manipulatives and activity sheets are provided. Extension/Follow Up: Authors: Students can use the Internet to research the cost analysis of this project. (i.e., fencing, cement, labor, etc.) John L. Hess, Jr. Stephen Decatur Middle School Prince George s County, MD Paula Ann S. Katze Bel Air High School Harford County, MD Gezell A. Montague McDonough High School Charles County, MD Jamese S. Shearin Eleanor Roosevelt High School Prince George s County, MD

3 Name: Day 1 Warm-up: This warm-up is intended to review the concepts of perimeter and area of a rectangle. DIRECTIONS: Calculate the perimeter and the area of each rectangle Perimeter = Area = Perimeter = Area = Perimeter = a Area = 4. A rectangle has an area of 2,130 and a width of 30 ; find its length and perimeter. 5. The perimeter of the triangle below is 52 cm. Find the length of each side of the triangle. Show your calculations. x

4 Name Activity 1: Can two rectangles with identical perimeters have different areas? Remember: Each paper square is 1 cm x 1 cm. You do not need to use all the paper squares. 1) a) Construct a rectangle which has a perimeter of 20 centimeters using the paper squares. After it is constructed, sketch your rectangle below and label the dimensions of the sides. b) What is the area of the rectangle you constructed? Show your calculations below. 2) a) Can you construct a second rectangle with the paper squares which still has a perimeter of 20 centimeters but whose area is different? If you are able to, sketch it below. b) What is the area of this rectangle? Show your calculations below. 3) a) If you were not restricted to using the paper squares, how many different rectangles with a perimeter of 20 centimeters do you think could be constructed?

5 b) Explain in paragraph form (2 to 3 sentences) how you came to your conclusion above. c) Will all the rectangles have identical areas? 4) Provide a written conclusion in sentence form which answers the original question posed at the top of the first page.

6 Activity 1 Materials: The blocks below are in sets of twenty five. Each student should receive one set, cut them out and use them to construct the rectangles discussed in the previous activity.

7 Name Day 2 Warm-up: This warm-up is intended to review how to calculate area and perimeter and refresh students on yesterday s concepts. 1. Calculate the area and perimeter of the rectangle: Perimeter = Area = 2. A rectangle has an area of 224 in.2 and a width of 16 in. a) Find the length of the rectangle. b) What is the perimeter of the rectangle? c) Sketch a rectangle below whose perimeter is the same as in (b) but whose area is different. Include dimensions in your sketch. 3. The length of a rectangle is 4 cm more than the width. The perimeter is 36 cm. Find the length and width of the rectangle. Show your calculations. 4. In paragraph form (3 or 4 sentences), explain what conclusions the class made yesterday about area and perimeter?

9 Your Property (for problem #1) Apple Avenue 52 Your House Your House I. Given the dimensions of your property, your task is to build an isolated pen (away from the house) for Penny behind your house. You want to have the largest possible area for the pen to provide Penny room to roam freely. Write the data numbers in the spaces above and sketch or draw Penny s pen on your property. (Remember to label all measurements.)

10 NAME PROBLEM #1 I. List the assumptions that are most relevant in determining the dimensions of Penny s pen. II. Fill in the table below and find a pattern to determine the most suitable dimensions for Penny s pen. [A] [B] [C] [D] [E] [F] [Input] 2*[A] 40-[B] [C]/2 [B] + [C] [A]*[D] LENGTH 2*LENGTH 2*WIDTH WIDTH PERIMETER PERIMETER AREA III. Based on above findings, what is the maximum area that Penny s pen can be? a) b)explain. c)what geometric shape is Penny s pen? IV. Use your straightedge to draw Penny s pen on your property. V. With your teacher, you will now construct the table you have created above on a graphing calculator. See Addendum 1 for detailed instructions.

11 Your Property (for problem #2) Apple Avenue 52 Your House Your House II. Your next task is to design a pen for Penny which is not necessarily isolated (away from the house). You still want to maximize the area. (Remember to label all measurements and make a new sketch.)

12 NAME PROBLEM #2 I. List the assumptions that are most relevant in determining the dimensions of Penny s pen (at least one should be different from those you listed in Problem #1). II. With your partner, design a table using a graphing calculator to solve the problem. Your table should make use of formulas to calculate data. When the table is complete, copy your column headings (the headings you would use if you were writing on paper), formulas, and the line of data which contains the maximum area in the table below. Graph. Calc. Headings -> Column Headings -> L1 L2 L3 L4 L5 L6 Formulas -> Maximum Area -> III. Based on the table you built, what is the new maximum area that Penny s pen can be? a) b) Explain. c) What geometric shape is Penny s pen? IV. Use your straightedge to draw Penny s pen on your property.

13 Your Property (with garage)(for problem #3) Apple Avenue Your House Your House 8 garage III. Your final task is to build a different pen for Penny. Remember you only have 40 feet of fencing to work with and you still want to have the largest possible area. (Remember to label all measurements.) (Hint: Your pen should not be isolated(away from the house).)

16 8) Arrow over to the L5 heading. This column calculates the perimeter of the fencing in feet. A formula you could use is (2*length) + (2*width). remember this is: L2 L4 To enter this formula, follow these keystrokes: 2nd ; 2 ; + ; 2nd ; 4 ; ENTER. 9) Arrow over to the L6 heading. This final column calculates area. A formula you could use is: length * width. Which two columns were these?. 10) Enter this formula by following these keystrokes: 2nd ; 1 ; x ; 2nd ; 3 ; ENTER 11) Your table is now complete and should match the table you constructed using paper and pencil in Problem #1.

17 Name: ANSWER KEY Day 1 Warm-up: This warm-up is intended to review the concepts of perimeter and area of a rectangle. DIRECTIONS: Calculate the perimeter and the area of each rectangle Perimeter = 58 Area = 204 SQ. FT Perimeter = 188 Area = 2088 SQ.FT Perimeter = a a Area = 24a SQ.FT. 4. A rectangle has an area of 2,130 and a width of 30 ; find its length and perimeter. LENGTH = 71 AND PERIMETER = The perimeter of the triangle below is 52 cm. Find the length of each side of the triangle. Show your calculations. x x + (2x+3) + (3x+1) = 52 cm 6x + 4 = 52 6x = 48 x = 8 cm 2x + 3 = 19 cm 3x +1 = 25 cm Triangle side lengths are: 8 cm,19 cm, and 25 cm

18 Name ANSWER KEY Student answers may vary and these are possibilities... Activity 1: Can two rectangles with identical perimeters have different areas? Remember: Each paper square is 1cm x 1cm. You do not need to use all the paper squares. 1) a) Construct a rectangle which has a perimeter of 20 centimeters using the paper squares. After it is constructed, sketch your rectangle below and label the dimensions of the sides b) What is the area of the rectangle you constructed? Show your calculations below. 6 x 4 = 24 cm 2 5 x 5 = 25 cm 2 2 x 8 = 16 cm 2 2) a) Can you construct a second rectangle with the paper squares which still has a perimeter of 20 centimeters but whose area is different? If you are able to, sketch it below. (Use one of the above rectangles) b) What is the area of this rectangle? Show your calculations below. (See one of the above) 3) a) If you were not restricted to using the paper squares, how many different rectangles with a perimeter of 20 centimeters do you think could be constructed? An infinite number b) Explain in paragraph form (2 to 3 sentences) how you came to your conclusion above. There were two rectangles that were formed earlier with blocks of length one. However, decimals could be used, as such: etc. c) Will all the rectangles have identical areas? No 4) Provide a written conclusion in sentence form which answers the original question posed at the top of the first page. Two rectangles with identical perimeters may have different areas.

19 Name ANSWER KEY Day 2 Warm-up: This warm-up is intended to review how to calculate area and perimeter and refresh students on yesterday s concepts. 1. Calculate the area and perimeter of the rectangle: Perimeter = 39 Area = 77 ft 2 2. A rectangle has an area of 224 in2 and a width of 16 in. a) Find the length of the rectangle. 224 / 16 = 14 in. b) What is the perimeter of the rectangle? (2 x14) + (2 x 16) = 60 in. c) Sketch a rectangle below whose perimeter is the same as in (b) but whose area is different. Include dimensions in your sketch. Possible answers A A = 200 in A= in The length of a rectangle is 4 cm more than the width. The perimeter is 36 cm. Find the length and width of the rectangle. Show your calculations. x + 4 x = width x + x + 4 = 36 width = 16 cm x x + 4 = length 2x + 4 = 36 length = 20 cm 2x = 32 x = 16 so x + 4 = In paragraph form (3 or 4 sentences), explain what conclusions the class made yesterday about area and perimeter? We concluded that if perimeter stays constant, area can change. We also were reminded that A=lw and P=2w +2l. We also practiced area and perimeter problems.

20 NAME ANSWER KEY PROBLEM #1 I. List the assumptions that are most relevant in determining the dimensions of Penny s pen. Penny s pen must be rectangular. Assume 40 feet of fencing. Penny s pen must be away from the house. Penny s pen must have the most area possible. II. Fill in the table below and find a pattern to determine the most suitable dimensions for Penny s pen. Some possible lines of data are: (correct answer is length 10) [A] [B] [C] [D] [E] [F] [Input] 2*[A] 40-[B] [C]/2 [B] + [C] [A] * [D] LENGTH 2*LENGTH 2*WIDTH WIDTH PERIMETER PERIMETER AREA III. Based on above findings, what is the maximum area that Penny s pen can be? a) 100 square feet b) Explain. All other lengths and widths yielded smaller areas than 100 square feet. The fencing used remained at 40 feet. c) What geometric shape is Penny s pen? 10 x 10 This is a square. IV. Use your straightedge to draw Penny s pen on your property. V. With your teacher, you will now construct the table you have created above on a graphing calculator. See Addendum 1 for detailed instructions.

21 ANSWER KEY Your Property (for problem #1) Apple Avenue Your House Your House PEN This is one possible location of pen. 124 I. Given the dimensions of your property, your task is to build an isolated pen (away from the house) for Penny behind your house. You want to have the largest possible area for the pen to provide Penny room to roam freely. Write the data numbers in the spaces above and sketch or draw Penny s pen on your property. (Remember to label all measurements.)

22 NAME ANSWER KEY PROBLEM #2 I. List the assumptions that are most relevant in determining the dimensions of Penny s pen (at least one should be different from those you listed in Problem #1). Penny s pen must be rectangular. Assume 40 feet of fencing. Penny s pen is not necessarily isolated. Penny s pen must have the most area possible. II. With your partner, design a table using a graphing calculator to solve the problem. Your table should make use of formulas to calculate data. When the table is complete, copy your column headings (the headings you would use if you were writing on paper), formulas, and the line of data which contains the maximum area in the table below. Graph. Calc. L1 L2 L3 L4 L5 Headings -> Column Headings -> -> Column Length 2*Length Width Fence Area Perimeter Formulas -> Data 2*L L2 L2 + L3 L1 x L3 Entry Maximum Area -> III. Based on the table you built, what is the new maximum area that Penny s pen can be? a) 200 square feet b) Explain. All other lengths and widths yielded smaller areas. The perimeter of fencing used remained at 40 feet. c) What geometric shape is Penny s pen? 10 x 20 This is a rectangle. IV. Use your straightedge to draw Penny s pen on your property.

23 ANSWER KEY Your Property (for problem #2) Apple Avenue Your House Your House PEN This is one possible location of pen. 124 II. Your next task is to design a pen for Penny which is not necessarily isolated (away from the house). You still want to maximize the area. (Remember to label all measurements and make a new sketch.)

25 ANSWER KEY Your Property (with garage)(for problem #3) Apple Avenue 40 Your House Your House garage 15 PEN III. Your final task is to build a different pen for Penny. Remember you only have 40 feet of fencing to work with and you still want to have the largest possible area. (Remember to label all measurements.) (Hint: Your pen should not be isolated(away from the house).)

27 7) A formula you could use is: (2*width) / 2 so this will be entered by following these keystrokes: 2nd ; 4 ; Divide ; 2 ; ENTER. 8) Arrow over to the L5 heading. This column calculates the perimeter of the fencing in feet. A formula you could use is (2*length) + (2*width). remember this is: L2 L4 To enter this formula, follow these keystrokes: 2nd ; 2 ; + ; 2nd ; 4 ; ENTER. 9) Arrow over to the L6 heading. This final column calculates area. A formula you could use is: length * width. Which two columns were these? L1 and L3 10) Enter this formula by following these keystrokes: 2nd ; 1 ; x ; 2nd ; 3 ; ENTER 11) Your table is now complete and should match the table you constructed using paper and pencil in Problem #1.

28 Performance Assessment Teacher s Guide: Introduction The purpose of the assessment activity is to provide feedback to tell you whether students learned the concepts that were taught. The information gathered from this assessment can be used so that appropriate instructional decisions can be made, such as whether to start a new unit or reteach. This assessment should be given at the conclusion of the learning unit on perimeter and area. There are a few questions designed to assess whether students can extend the concept of perimeter to triangles. The questions are centered around real-world applications as much as possible. Objectives - Students will demonstrate their ability to: use the formulas for perimeter and area of a rectangle to solve problems. solve one and two step equations. translate word sentences into algebraic expressions. use appropriate units to describe perimeter and area. Tools/Materials Needed for Assessment Pencil Calculator Straightedge Administering the Assessment Students should complete the assessment individually. They may write directly on the assessment sheet. Students should be able to complete this assessment in approximately 30 minutes.

29 Performance Assessment Student Response Sheet Perimeter and Area Assessment Task Name: Date: For problems 1-3, select the best response. 1) If you know the area of a rectangle and its length, to find its width you need to a. add b. subtract c. multiply d. divide 2) The area of a rectangle describes its a. distance b. surface c. angles d. sides 3) If the length of a rectangle is cut in half but its width is doubled, then the area of the rectangle would a. double b. halve c. quadruple d. remain the same 4) The of a triangle is the sum of the lengths of its sides. 5) The geometric figure that maximizes area is a. 6) The cost of carpet is \$3 per square foot. How much would it cost to carpet a bedroom that is 11 ft. by 14 ft.? Show all steps, include formula used and proper units. 7) A rectangular picture frame has a perimeter of 6 ft. The width of the frame is 0.80 times its height. Find the height of the frame in inches. Show all steps, include diagram, formulas used, and proper units.

30 8a) A student types a book report on a 8 in. by 11 in. piece of paper. The top and bottom margins are set at 1.5 in. each and the left and right margins are set at 1 in. each. He did not indent his paragraphs and he used the entire page to type, excluding the margins. What percentage of the page did he use for typing? Show all steps, include diagram, formulas used, proper units, and answer in a complete sentence. 8b) The student s completed book report takes up one full page. He was very careful not to write in the margins of the paper. He would like his report to stand out, so he plans to write his final draft on a piece of paper with unusual dimensions and no margins. Design a piece of paper that has the exact amount of area the student will need, but whose dimensions are different from the space he wrote in previously. What is the perimeter of his new piece of paper? Show all work, include diagrams, and write your answer in a complete sentence.

31 Performance Assessment Scoring Guide Perimeter and Area Assessment Task Scoring Guide Name: Date: For problems 1-3, select the best response. 1) If you know the area of a rectangle and its length, to find its width you need to a. add b. subtract c. multiply d. divide 2) The area of a rectangle describes its a. distance b. surface c. angles d. sides 3) If the length of a rectangle is cut in half but its width is doubled, then the area of the rectangle would a. double b. halve c. quadruple d. remain the same 4) The perimeter of a triangle is the sum of the lengths of its sides. 5) The geometric figure that maximizes area is a square. 6) The cost of carpet is \$3 per square foot. How much would it cost to carpet a bedroom that is 11 ft. by 14 ft.? Show all steps, include formula used and proper units. \$ The student uses area formula correctly with proper units and calculates the cost of the carpet. 2 - The student uses area formula correctly without proper units and calculates the cost of the carpet. 1 - The student uses area formula incorrectly without proper units and calculates the cost of the carpet incorrectly. 0 - no response

32 7) A rectangular picture frame has a perimeter of 6 ft. The width of the frame is 0.80 times its height. Find the height of the frame in inches. Show all steps, include diagram, formulas used, and proper units.20 in 3 - The student uses perimeter formula correctly with proper units, includes diagram, and calculates height correctly with proper units. 2 - The student uses perimeter formula correctly with proper units, includes diagram, and calculates height correctly without proper units. 1 - The student uses perimeter formula incorrectly, resulting in an incorrect answer. 0 - no response 8a) A student types a book report on a 8 in. by 11 in. piece of paper. The top and bottom margins are set at 1.5 in. each and the left and right margins are set at 1 in. each. He did not indent his paragraphs and he used the entire page to type, excluding the margins. What percentage of the page did he use for typing? Show all steps, include diagram, formulas used, proper units, and answer in a complete sentence % 4 - The student uses area formula correctly with proper units, includes diagram, calculates percentage correctly, and answers in a complete sentence. 3 - Student makes one of the following errors: uses area formula incorrectly, does not make a diagram; makes the diagram incorrectly, calculates percentage incorrectly; does not use proper units, or does not answer in a complete sentence. 2 - Student makes two of the following errors: uses area formula incorrectly, does not make a diagram; makes the diagram incorrectly, calculates percentage incorrectly, does not use proper units, or does not answer in a complete sentence. 1 - Student makes three of the following errors: uses area formula incorrectly, does not make a diagram, makes the diagram incorrectly, calculates percentage incorrectly, does not use proper units, or does not answer in a complete sentence. 0 - No response

33 8b) The student s completed book report takes up one full page. He was very careful not to write in the margins of the paper. He would like his report to stand out, so he plans to write his final draft on a piece of paper with unusual dimensions and no margins. Design a piece of paper that has the exact amount of area the student will need, but whose dimensions are different from the space he wrote in previously. What is the perimeter of his new piece of paper? Show all work, include diagrams, and write your answer in a complete sentence. (perimeter will vary) 4 - The student makes a diagram, chosen perimeter corresponds to correct area of 48 in2, shows calculations, and writes answer in a complete sentence. 3 - Student makes one of the following errors: does not make a diagram, chosen perimeter does not correspond to correct area of 48 in2, does not show calculations, or does not write answer in a complete sentence. 2 - Student makes two of the following errors: does not make a diagram, chosen perimeter does not correspond to correct area of 48 in2, does not show calculations, or does not write answer in a complete sentence. 1 - Student makes three of the following errors: does not make a diagram, chosen perimeter does not correspond to correct area of 48 in2, does not show calculations, or does not write answer in a complete sentence. 0 - No response

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