Students will: Derive the formula for the area of a parallelogram. Relate the area of a parallelogram to the area of a triangle.

Transcription

1 Lesson13 AREA Linear Measurement and LESSON OBJECTIVES Students wi: erive the formua for the area of a paraeogram. Reate the area of a paraeogram to the area of a triange. PREREQUISITES Students shoud be abe to: Recognize rectanges and trianges. Cacuate the area of a rectange. RELATE STAMS LESSONS Book Lesson 14 Understand introduces students to the concept of area. Book Lesson 15 of Rectanges has students use mutipication and formuas to find the area of a rectange. MATH BACKGROUN In this esson, students earn how to find the area of paraeograms and trianges. Students are shown how the famiiar formua for finding the area of a rectange, A 5 3 w, can be used to find the area of a paraeogram, A 5 b 3 h. The base and height of a paraeogram correspond to the ength and width of a rectange. A rectange is actuay a paraeogram with four right anges, which is why the ength and width can be used as the base and height. In Part Two, students see that any triange has the same area as haf of a paraeogram with the same base and height. Showing students how shapes compare can hep them remember based on their understanding, rather than by trying to memorize many formuas. Encourage students to think about area in the word around them. For exampe, finding how much materia is needed for a tent might incude finding the area of trianges. VOCABULARY Page 14 area: the amount of space a two-dimensiona figure covers formua: a mathematica statement that uses numbers and symbos to describe a quantity paraeogram: a quadriatera with two pairs of parae sides parae: a description of ines that never intersect and aways remain the same distance apart Page 15 right ange: an ange that measures 90 degrees Page 16 triange: a poygon with three sides congruent: having the same size and shape Interactive Whiteboard Visuaize of Paraeograms and Trianges Go to the Interactive Whiteboard Lessons to bring Parts One and Two to ife. Use features such as manipuabe modes to deepen students understanding of finding the area of paraeograms and trianges. 130 Geometry and Measurement

2 Modeed Instruction Lesson Guided Instruction 13 AREA Think It Through PART ONE: Learn About of a Paraeogram? Fi in the banks. Sove the probem. How can you use a rectange to find the area of a paraeogram? Gino is going to pave his patio. It is in the shape of a paraeogram. (A) tes how much surface a two-dimensiona figure covers. You can use the formua A 5 3 w to find the area of a rectange. How can you find the area of a paraeogram? 8 ft 10 ft What is the area Gino needs to pave? A paraeogram has four sides. The opposite sides are parae. They are the same distance apart and never meet. height width base feet and h 5 8 feet. In the paraeogram, b 5 10 Use the formua for the area of a paraeogram. A 5 b 3 h ength Paraeogram Rectange If you cut a triange off of one end and move it to the other end, you get a rectange. In the paraeogram, b 5 4 units and h 5 3 units. In the rectange, 5 Find the dimensions. The base and height of a paraeogram aways form a right ange. Sometimes the height of the paraeogram is not one of the sides. 3 8 A 5 10 square feet A 5 80 square feet of his patio. Soution: Gino needs to pave 80 4 units and w 5 3 units. Now, use what you know to sove this probem. The paraeogram and rectange cover the same area on the grid. Instead of A 5 3 w, you can use A 5 b 3 h to find the area of a paraeogram. Figure imensions Formua A 5 b 3 h Paraeogram b and h A 5 b 3 h A A 5 1 square units Rectange and w A 5 3 w 1. What is the area of this paraeogram? 4 ft 5 ft A 1 sq ft B 9 sq ft C 13 sq ft If you know the area of a paraeogram and its base, how can you find its height? Expain. 14 Students activate their background knowedge about finding the area of a rectange to earn to find the area of a paraeogram. Page 14 Introduce the Question at the top of the page. In Expore, have students draw a rectange and abe ength and width. Make sure students can describe area in their own words. Read Think with students. Hep students identify the properties of a paraeogram. Compare with those of a rectange. Tip: raw other paraeograms so students can see that they do not aways ook the same. Use the grid to show that the space covered by the rectange and the paraeogram is the same. iscuss Connect with students. Hep students identify the vaues for the base and height of the paraeogram. Organize students in pairs or groups for Let s Tak and monitor their discussions. Geometry and Measurement 0 sq ft 15 Be sure students understand that, since the area is the product of the base and the height, you can divide the area by the base to find the height. Page 15 Read the Think It Through probem with students. Hep students determine that they coud redraw the paraeogram as a rectange. Show students they coud use the formua for area of a rectange to find the area of the paraeogram. ELL Support: The word right aso means correct. Carify that in math a right ange has a measure of 90 degrees. Monitor students as they compete Your Turn. Then discuss the correct answer. Error Aert: Students who chose B may not have paid attention to the operation sign, 3, in the area formua. See Hands-on Activity (page 137). 131

3 Modeed Instruction Guided Instruction PART TWO: Learn About of a Triange? Think It Through How can you find the area of a triange? You can use the formua A 5 b 3 h to find the area of a paraeogram. How can you use the area of a paraeogram to find the area of a triange? Fi in the banks. Sove the probem. Aexis is making a garden in the shape of a triange. 6m A triange is a poygon with three sides. If you draw a second triange with the same size and shape, you can form a paraeogram. The trianges are congruent. b h b h h b The base of each triange is 10 units. The height of each triange is 4 units. What fraction of the paraeogram is taken up by each _ 1 triange? Gr5_SB_L13_art_04.eps 9m What is the area of her garden? Aexis can draw a second triange that is congruent to her garden. She can arrange the trianges into a paraeogram. Just ike the paraeogram, the base and The height of the paraeogram is 6 meters. The base of the paraeogram is 9 meters. height of the triange must form a right ange. In this case, the height the area of the paraeogram. The area of the triange is haf The area of the paraeogram is A 5 b 3 h, so the area of the triange is A 5 1 _ (b 3 h). of the triange is a side of the triange. Use the formua to find the area. The paraeogram and the two trianges cover the same area on the grid. The area of the paraeogram is A 5 b 3 h, so the area of one triange must be 1 (b 3 h). The base and height of each triange are the base and height of the paraeogram. 1 (b 3 h) 1 (10 3 4) 1 (40) 5 0 square units 1 _ A 5 ( ) A 5 7 square meters. Soution: The area of Aexis s garden is 7 Now, use what you know to sove this probem.. What is the area of this triange? Use the formua. 10 cm How can you check the exampe to prove that the area of each triange is 1 the area of the paraeogram? Use your method to show that the answer is correct. 16 Students earn that they can use the area of a paraeogram to find the area of a triange. Page 16 Introduce the Question at the top of the page. Read Expore with students. Reinforce that students can find the area of a paraeogram by mutipying the base by the height. ELL Support: The prefix tri- means three. Review some words with this prefix, such as tricyce. Point out that knowing this prefix wi hep students remember that a triange has three sides. Read Think with students. Pause so students can read aoud the responses in green. Hep students identify the vaues for the base and height of the triange for Connect. Guide students through the computation. Organize students in pairs or groups for Let s Tak and monitor their discussions cm 1 _ A 5 ( ) Gr5_SB_L13_art_06.eps sq cm A Be sure students understand that they can doube the area of one triange. Then they can compare that amount with the area of the paraeogram to see that the amounts are the same. Page 17 Read the Think It Through probem with students. Guide students as they sove the probem. Pause for students to fi in missing information. Then discuss each response. Tip: Have students sketch a triange so that the garden and the new triange create a paraeogram. Point out that since the paraeogram has 4 right anges, it is a rectange. Monitor students as they compete Your Turn. Then discuss the correct answer. Error Aert: Students who wrote an incorrect formua may be confused with the area formua for a paraeogram. See Reteaching Activity (page 137). Geometry and Measurement

4 Modeed Practice Guided Practice PART THREE: Choose the Right Answer Sove each probem. Use the hints to avoid mistakes. Sove the probem. Then read why each answer choice is correct or not correct. Choose the correct formua to find the area of a paraeogram or a triange. I dentify the measurements for the base and height of the figure. Mr. Sanchez is painting shapes on his son s wa. One shape is this triange. R epace the variabes in the formua with the measurements and sove. 10 cm 4 cm 3. Oivia drew this paraeogram. 9 cm Which equation shows how to find the area of the triange? 5. A restaurant has a tabe in the shape of a triange. 5 in. A in. 1 in. Gr5_SB_L13_art_07.eps B Which expression can you use to find the area of the paraeogram? C _ in. Which expression can you use to find Gr5_SB_L13_art_10.eps the area of the tabe? A B _ 1 (1 3 5) A Check to see if you chose the correct answer. The base of the triange is 9 centimeters, and the height is 4 centimeters. 1 (b 3 h) 1 (9 3 4) 5 18 So, the correct answer is. C _ 1 (30 1 0) A triange has a height of 14 inches and a base of 8 inches. What is the area of the triange? 6 ft B The product of base and height must be mutipied by _ 1. C This is the perimeter of the triange. Use the formua for area A store ses a carpet in the shape of a paraeogram. 10 ft This is the sum of the base and height. To find the area, mutipy _ 1 by the product of the base and height. B Why are the other answer choices not correct? A _ 1 (30 3 0) C A 11 sq in. B sq in. What is the area of the carpet? C A 16 sq ft Gr5_SB_L13_art_09.eps B 30 sq ft C 56 sq in. 11 sq in. 60 sq ft 106 sq ft 19 Answer Anaysis Students reinforce their understanding of area through soving a mutipe-choice probem and anayzing correct and incorrect answer choices. 3. A Added twice the base and twice the height B Used the formua for a triange C Added instead of mutipied The area of a paraeogram is A 5 b 3 h. Page 18 Te students that this page modes finding the correct answer to a mutipe-choice probem. Have students read the probem in Sove and choose the best answer. Remind students to check their math. Examine Check with students. iscuss the correct and incorrect choices. Page 19 Monitor students as they compete Your Turn. Organize students in pairs or sma groups and have them discuss why each answer choice is correct or not and what errors may have been made. Review the answers with the cass. 4. A Added the base and the height B Mutipied by _ 1 A 5 b 3 h Put the numbers of the base and height together instead of finding their product 5. B C The area of a triange is _ 1 (b 3 h). Used the formua for the area of a paraeogram Added the base to the height instead of mutipied id not use the correct formua for the area of a triange 6. A Added the base and the height before 1 mutipying by _ B Found the sum of the base and the height A 5 _ 1 (b 3 h) 5 _ 1 (8 3 14) 5 56 See Vocabuary Activity (page 137). Forgot to mutipy by _ 1 Geometry and Measurement 133

5 Modeed Practice Guided Practice PART FOUR: Write the Best Answer Sove the probem. Use what you earned from the mode. Study the mode. It is a good exampe of a written answer. CHECKLIST 7. Pedro drew this shape on a poster. What is the area of the shape? Student Mode id you... show each step? Riey is panting grass seed in the yard shown in the picture. What is the area of the yard that Riey is panting? answer the question 7 in. asked? 10 in. 6 yd Use pictures, words, or numbers to show your work. 1 yd Use pictures, words, or numbers to show your work. The student shows each step. A5b3h b 5 1 and h 5 6 A A 5 7 Soution: 7 The student correcty answers the question asked. square yards Expain how you got your answer. Because the shape of Riey s yard is a paraeogram, I used the formua A 5 b 3 h. The base is 1 yards and the height is 6 yards. I put those vaues into the formua and then found the area: So, the area is 7 square yards. 130 give important detais? The student gives important detais about how to find the area. use math words? A5b3h b 5 10 and h 5 7 A A 5 70 in. Soution: 70 square inches Expain how you got your answer. Because the shape on Pedro s poster is a paraeogram, I used the formua A 5 b 3 h. The base is 10 inches, and the height is 7 inches. I put those vaues into the formua and then found the area: So, the area is 70 square inches. The student uses the math words paraeogram, formua, base, height, and area. Students study a mode answer to an extended-response probem. Page 130 Te students that this page modes buiding the soution to a probem one step at a time and writing to expain the soution. Have students read the probem in Show. iscuss how the drawing and each mathematica step ead to the soution. Tip: Expain that different paces and objects, for exampe foor ties and garden pots, can have the shape of a paraeogram. No matter what the paraeogram describes, the area is found in the same way. Read Expain with students. Have students circe the math words in the expanation. 131 irect students attention to the notes in the right margin. Te students that this mode woud receive a high score for the reasons described in these notes. Page 131 Monitor students as they compete Your Turn. Encourage students to foow the Checkist to write the best answer. Have students discuss their work with a partner. Then discuss the correct answer as a cass. Answer and Expanation 7. See the sampe answer. This answer shows a of the steps taken to sove the probem. The soution answers the question. The expanation provides important detais about how the probem was soved and uses the math words paraeogram, formua, base, height, and area. See Rea-Word Connection (page 137). See Schoo-Home Connection (page 137). 134 Geometry and Measurement

6 Independent Practice Independent Practice PART FIVE: Prepare for a Test 1. Mrs. Ramirez drew this shape with chak on the payground. As you sove area probems, remember to 4 ft draw a picture of the figure if one is not given. 15 ft write the formua you wi use. Then sove the probem. check your work by using division. A 10. Which paraeogram has an area of 18 square units? 8 units 5 units 10 units B A B C units 8 in. 9 uints 10 in. B C (10 3 8) A 48 8 yards 1 yards square yards A 5 _1 (b 3 h). The base is 1 yards and the height is 8 yards. I put those vaues into the formua and then found the area. I mutipied to find that 3 cm B 6 cm 1 (1 3 8) 5 48, so the area is 48 square yards. Students practice using area formuas to sove probems that might appear on a mathematics test. Pages Te students that they wi practice soving probems that invove the areas of paraeograms and trianges. Point out the tips at the top of page 13. Expain to students that these tips wi hep them answer the probems correcty. You may wish to have students review the hints for avoiding mistakes on page 19 as we. Te students to compete probems 8 15 on pages 13 and 133. Encourage students to check their answers. iscuss the correct responses as a cass. Geometry and Measurement A 5 15 square feet I drew a picture of the yard. Because the shape is a triange, I used the formua 1 cm 30 sq yd A 5 _ Expain how you found your answer. C 8 cm 13 C 3 sq yd 15 sq yd Soution: 11. A paraeogram has a base of 8 centimeters. Its area is 4 square centimeters. What is the height of the paraeogram? 1 _ A 13 sq yd B A 5 _1 (b 3 h) A 5 _1 (1 3 8) A 5 48 sq yd 9 units A Write an equation to find the area of Gr5_SB_L13_art_0.eps the front of the tent. Use pictures, words, or numbers to show your work. 4 units 9. Which equation can be used to find the area of the triange? 11 sq ft 15. Joshua s front yard forms a triange. The base of the triange is 1 yards. The height of the triange is 8 yards. What is the area of Joshua s front yard? 6 units C 19 sq ft 3 units C sq ft B 30 sq ft 6 ft 13. A marching band is forming a triange with a base of 10 yards and a height of 3 yards. What is the area of the triange? A 3 units 5 ft What is the area of the shape? Sove each probem. 8. Which expression can be used to find the area of the paraeogram? 14. Tyrone s famiy is going camping. The picture shows their tent. 133 Answers and Expanations 8. The base and height are provided in the drawing. A5b3h C The area of a triange is one-haf the base times the height. A 5 _ 1 (b 3 h) 5 _ 1 (10 3 8) B Each paraeogram has a different area. The area of this paraeogram is 18 square units. A 5 b 3 h A A 5 b 3 h. If A 5 4, 8 3 h must equa 4. So the height must be 3 centimeters. 1. A The base is 15 feet, and the height is 4 feet. A 5 b 3 h B A 5 1 (b 3 h) 5 1 (10 3 3) 5 15 (continued on page 136) 135

7 (continued from page 135) 14. A 5 1 (5 3 6) 5 1 (30) 5 15; the area of the front of the tent is 15 square feet. 15. See the sampe answer. This answer shows a of the steps the student took to sove the probem. The soution answers the question. The expanation provides important detais about how the student soved the probem and uses the math words formua, base, height, triange, and area. ASSESSMENT AN REMEIATION Ask students to find the area of a triange with a base of 9 feet and a height of 6 feet. (7 sq ft) For students who are sti strugging, use the chart beow to guide remediation. After providing remediation, check students understanding. Ask students to expain their thinking whie finding the area of a triange with a base of 7 inches and a height of 1 inches. (4 sq in.) If a student is sti having difficuty, use STAMS Book, Lesson 15, pages If the error is... Students may... To remediate... the formua is not correct the area is twice what it shoud be have confused a different formua for the area of a triange not understand how a paraeogram can be divided into two congruent trianges. Encourage students to draw a rectange, a paraeogram, and a triange. Then have them abe the formua for finding the area of each figure. Provide grid paper to have students practice reating the area of congruent trianges to the area of a paraeogram. the cacuation is incorrect the perimeter is found instead of the area have mutipied by rather Have students gather a group of items. Ask them to than 1, or separate the items into two groups. Show them that this is what they do when they mutipy by 1. have mutipied both the base and the height by 1 and then found the sum. be confusing addition with mutipication, or Review the formua for the area of a triange. Show students that they can mutipy the base by the height first, and then mutipy the product by 1. Use grid paper to show students how to find the area of a rectange by mutipying the number of squares in each row by the number of rows. have misabeed the base and height of a triange. raw a triange on the board. Point out the base, the height, and the sides. For students who have mastered the skis in this esson, see Chaenge Activity (page 137). 136 Geometry and Measurement

8 AITIONAL ACTIVITIES Hands-on Activity Use paper to reate rectanges to paraeograms. Materias: sheet of coored paper, ruer, scissors, penci Organize students in pairs and distribute paper, ruers, scissors, and pencis. Have students draw a paraeogram of any size on the paper. Ask students to cut out the paraeogram and discuss the characteristics of the figure. Then te students to draw a ine that forms a right triange on one end of the paraeogram. Have students cut aong the ine. Students shoud then move the triange to change the paraeogram into a rectange. Ask students to expain how they know that both shapes cover the same area. Encourage different groups to compare their figures and show that a paraeograms can be changed into rectanges that cover the same area. Reteaching Activity Use various grids to mode area. Materias: grid paper with arge squares, crayons, scissors istribute paper, crayons, and scissors to each student. Te students to draw a triange on the grid paper and coor it. Then have them cut out the triange, trace it, and cut out another triange. Point out that the trianges are the same size and shape. irect students to pace the trianges side by side to form a paraeogram on the grid paper. Ask students, How many trianges did you make? () How is the area of the two trianges reated to the area of the paraeogram? (The area is the same.) How can you find the area of just one of the trianges? (ivide the area of the paraeogram by or mutipy by 1.) Ask vounteers to write the formua for finding the area of a paraeogram and then show how to mutipy the formua by 1. Vocabuary Activity Pay Bingo to reinforce terms. Materias: bank sheets of paper, counters Have each student create a grid by foding a sheet of paper in thirds horizontay and then in thirds verticay. ispay the vocabuary words, aong with the term square units. Then te students to write BINGO in the center box and the vocabuary words in the other boxes. Read a definition and have students cover the corresponding word on their grid with a counter. The winner for each round is the first student to cover 3 spaces verticay, horizontay, or diagonay. Rea-Word Connection Identify everyday exampes of area. Ask students to draw pictures to show exampes in which it is important to know area. For exampe, a person might be covering a foor with carpet or buying paint for a wa. Chaenge students to incude exampes of paraeograms and trianges, such as buying fabric to make a trianguar pennant. ispay their pictures. Schoo-Home Connection Inform famiies about area. Give each student a copy of the reproducibe Schoo-Home Connection for Lesson 13 (page 187) to share with the famiy. The activity in the etter has the famiy draw and determine the areas of a paraeogram and a triange. Chaenge Activity Create and describe designs. Have students create a design using paraeograms and rectanges. Students might cut identica figures and paste them into pace or draw a variety of figures. Then te students to measure the base and height of each figure and use the measurements to cacuate the tota area of the design. Geometry and Measurement 137