Algebra Area of Triangles

Transcription

1 LESSON 0.3 Algera Area of Triangles FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards Learning Ojective 6.G.A. Find te area of rigt triangles, oter triangles, special quadrilaterals, and polygons y composing into rectangles or decomposing into triangles and oter sapes; apply tese tecniques in te context of solving real-world and matematical prolems. Find te area of triangles. Also 6.EE.A.c Students write instructions for tutoring a younger student on ow to find te area of triangles. MATHEMATICAL PRACTICES (See Matematical Practices in GO Mat! in te Planning Guide for full text.) MP Make sense of prolems and persevere in solving tem. MP5 Use appropriate tools strategically. MP8 Look for and express regularity in repeated reasoning. F C R Coerence: Standards Across te Grades Before Grade 6 After 5.NF.B.4 6.G.A. 7.G.A. 7.G.B.6 F C R Rigor: Level : Understand Concepts...Sare and Sow ( Cecked Items) Level : Procedural Skills and Fluency...On Your Own, Practice and Homework Level 3: Applications...Tink Smarter and Go Deeper Language Ojective Materials MatBoard, Grid Paper (see eteacer Resources), scissors, straigtedge, color pencils FC R For more aout ow GO Mat! fosters Coerence witin te Content Standards and Matematical Progressions for tis capter, see page 53J. Aout te Mat Professional Development Using Models to Derive Formulas In tis capter, students use formulas to find te areas of a variety of geometric figures. Drawings can elp students see ow te formulas are derived and te relationsips among te different figures. In tis lesson, students learn aout te area formula for triangles. Students use previous knowledge regarding te area of a parallelogram to derive te area formulas for triangles. Students use grid paper to draw, manipulate, and compare figures wit given dimensions. For example, te model at te rigt sows two congruent triangles rearranged to form a parallelogram. Te corresponding area formula is given elow te model. Notice tat te two congruent triangles form a parallelogram and, terefore, te area of eac triangle is alf te area of te parallelogram. Professional Development Videos 545A Capter 0 Area of a triangle: A =

2 Daily Routines Common Core Prolem of te Day 0.3 Te ig scool is renovating its rectangular swimming pool. Tiles will cover te ottom of te 75 ft y 30 ft pool. Tey will not cover te 0 ft y 8 ft rectangular logo in te center of te floor. Determine te area of te pool floor tat needs tiles.,70 square feet Vocaulary Interactive Student Edition Multimedia Glossary e 34 Fluency Builder Materials eteacer Resources page TR34 Operations Find te quotient.. 3, , , , , Common Core Fluency Standard 6.NS.B. ENGAGE wit te Interactive Student Edition Essential Question How can you find te area of triangles? Making Connections Review te fact tat te area of a triangle equals _ te area of a parallelogram wit te same ase and eigt. Wat is true aout te ase and eigt of tis triangle and parallelogram? Tey are equal. Learning Activity Review te formulas for parallelogram and triangle area. Wat is te formula for te area of a parallelogram? A = were is te ase and is te eigt. Wat is te formula for te area of a triangle? A = _ were is te ase and is te eigt. How many variales does tis formula ave? 3 Literacy and Matematics Visit te lesson opener wit te students. Ten, ave students rewrite te prolem in two sentences or less. Have students write teir own real-world prolems tat involve finding te area of a triangle. 34 Pages in Strategies and Practice for Skills and Facts Fluency provide additional fluency support for tis lesson. How can you find te area of triangles? Lesson B

3 LESSON 0.3 EXPLORE Present te formula for te area of a triangle. Students may demonstrate using te formula to find te area of several small triangles, and ten confirm teir results using i Tools Geometry. Unlock te Prolem MATHEMATICAL PRACTICES Guide students to use te formula for te area of a triangle to solve te prolem. Wat information do you need to find te area of a triangle? You need te lengt of te ase and te eigt. Wy do you multiply y _? Te area of a parallelogram is found y multiplying te ase times te eigt. Te area of a triangle is alf te area of a parallelogram wit te same ase and eigt. Mat Talk 545 Capter 0 Use Mat Talk to focus on students understanding of areas of triangles. MP6 Attend to precision. How is te area formula for a triangle like te area formula for a parallelogram? How is it different? Possile answer: Bot formulas involve te product of te ase and eigt of te figure. Te formula for te triangle, owever divides tis product y. ELL Strategy: Scaffold Language Draw a rigt triangle and a parallelogram on a grid. Have students identify te sapes. Write te sentence frames and elp students complete tem. Guide students to descrie te sapes using te terms ase, eigt, and area. Te of te triangle is _. Te of te parallelogram is _. DEEPER MP Reason astractly and quantitatively. Would you get te same result if you multiplied 97.6 y _ and ten multiplied te product y 79? Explain. Yes; Possile explanation: Since multiplication is commutative, te answer will e te same even if te numers are multiplied in a different order. 6.G.A. Find te area of rigt triangles, oter triangles, special quadrilaterals, and polygons y composing into rectangles or decomposing into triangles and oter sapes; apply tese tecniques in te context of solving real-world and matematical prolems. Hougton Mifflin Harcourt Pulising Company Image Credits: (r) Frank Witney/Getty Images Algera Area of Triangles To find te area of a triangle, use te formula A 5 _ 3 ase 3 eigt. Find te area of te triangle. Find te area of te triangle.. Write te formula. A 5 _ 3. A = Area of Triangles Essential Question How can you find te area of triangles? Any parallelogram can e divided into two congruent triangles. Te area of eac triangle is alf te area of te parallelogram, so te area of a triangle is alf te product of its ase and its eigt. 7 ft 6 ft 6.5 m 4 m 3. ase Step Write te formula. A 5 _ Step Rewrite te formula. Sustitute te ase and eigt measurements for and. eigt 7 cm A 5 _ Step 3 Simplify y multiplying. A 5 _ 3 A Step 4 Use te appropriate units. A cm 3 m Capter Resources Hougton Mifflin Harcourt Pulising Company Unlock te Prolem Te Flatiron Building in New York is well known for its unusual sape. Te uilding was designed to fit te triangular plot of land formed y nd Street, Broadway, and Fift Avenue. Te diagram sows te dimensions of te triangular foundation of te uilding. Wat is te area of te triangle? 6 Sustitute for and. A 5 _ 3 3 Simplify. A = _ 3 0 A = 5 ft 97.6 ft Find te area of te triangle. Write te formula. Sustitute 97.6 for and 79 for. Multiply te ase and eigt. Multiply y _. Area of a Triangle were is te ase and is te eigt Prolem Type: Area Unknown Product 3 cm in. 9 in in. A = 0-9 Reteac 3 Differentiated Enric 0.3 Instruction Lesson 0.3 Reteac 79 ft ,60.4 A = 7,805. So, te area of te triangle is 7,805. ft. Reteac 0.3 Te area of a triangle is alf te area of a rectangle wit te same ase and eigt. Wordy Triangles Make a drawing to sow te situation. Lael te dimensions. Ten use te drawing to find te area.. A model sip as a mast tat is 7 inces tall. A rigt triangular sail goes from te top of te mast to inc from te ottom of te mast. Te lengt of te ase of te sail is 4 inces. Te eigt of te sail is along te mast. Area of sail: in. 6 in. 4 in. 7 in. in. 3. Andrew is elping te first-grade class make paper trees. For eac student, e folds a rectangular piece of construction paper in alf lengtwise. Ten, e cuts te folded paper into a triangular sape wit te same ase and eigt as te rectangle. Ten e unfolds te paper and trows away te scraps. Te original piece of paper is inces y 8.5 inces. in.. Two ropes extend from te top of a flagpole to te ground 8 feet away from te flagpole s ase on eiter side. Te flagpole is 30 feet ig. Te ropes eac create a rigt triangle wit te flagpole, and eac rigt triangle is alf of a larger triangle wit te flagpole at its center. Area of larger triangle: 4. Lucy s rectangular garden is 5 feet long and feet wide. Se adds a rigt triangle to er garden plan, so tat te triangle s eigt extends out 7 feet in line wit te sorter side of te rectangle, and te ase of te triangle runs from te corner of te rectangle to te center of te longer side of te rectangle. 7.5 ft 8.5 in. Area of paper trown away per student: Total area of Lucy s garden: Capter Resources Hougton Mifflin Harcourt Pulising Company in. ALGEBRA Lesson 0.3 Geometry 6.G.A. Also 6.EE.A.c MATHEMATICAL PRACTICES MP3, MP6, MP8 How can you identify te ase and te eigt of te triangle? Possile answer: Te ase is one of te sides of te triangle and te eigt forms a 90º angle wit te ase. Mat Talk MATHEMATICAL PRACTICES 8 Generalize How does te area of a triangle relate to te area of a rectangle wit te same ase and eigt? Capter ft 8 ft 30 ft ft Lesson 0.3 Enric Possile drawings are sown. 8 ft 06.5 ft 540 ft 5 ft 0-0 Enric

4 Example Write te formula. Sustitute 4 _ for and 3 _ for. A Rewrite te mixed numers as fractions. Multiply. Rewrite te fraction as a mixed numer. Example? in. A 5 5 in. 30 in. = 7 So, te area of te triangle is 7_ 8 ft. Write te formula. Sustitute 5 for A and 30 for. Multiply _ and _ _ 63 A = 8 7 A = 7_ 8 5 = 30 = 5 Use te Division Property of Equality. _ 5 5 = 5 5 Simplify. Find te area of te triangle. Daniella is decorating a triangular pennant for er wall. Te area of te pennant is 5 in. and te ase measures 30 in. Wat is te eigt of te triangular pennant? 5 = So, te eigt of te triangular pennant is 5 in. 4 _ 3 _ 3 ft 4 ft Prolem Type: Area Unknown Factor Hougton Mifflin Harcourt Pulising Company Image Credits: (r) Hougton Mifflin Harcourt Example Guide students troug Example. Review te formula for area of a triangle as well as ow to convert mixed numers to fractions greater tan one. Wy are te mixed numers rewritten as fractions greater tan one? Possile answer: We cange tem so we can multiply numerator y numerator and denominator y denominator. Example In tis example, te area of te triangle is given and students will need to find te eigt. Tis requires students to solve te equation for te eigt, not just evaluate te expression as tey ave done up to tis point. Wat information are you given in tis exercise? Wat are you asked to find? ase and area; eigt Is tere anoter way you could ave solved te prolem esides te steps given? Yes; I could ave divided 5 y 30 and ten multiplied ot sides y. 546 Advanced Learners Visual, Logical / Matematical Individual / Partners Materials Grid Paper (see eteacer Resources), color pencils Have students make composite figures on grid paper y drawing adjacent triangles and parallelograms. Have tem find te area of eac sape and ten te total area. Instruct students to erase interior lines or cover tem y coloring te entire new figure. Students sould excange figures wit teir partners and callenge eac oter to find te areas of te composite figures. Ceck answers for accuracy. Students will need to reak te figure into individual sapes, find te area of eac, and ten add tese measurements togeter. Te sum is equal to te total area. COMMON ERRORS Error Students may forget to multiply y _ wen finding te area of a triangle. Example A = _ 8 6 A = 48 Springoard to Learning Review te formula for te area of a triangle. Empasize te relationsip etween te area of a triangle and te area of a parallelogram wit te same ase and eigt. Lesson

5 3 EXPLAIN Sare and Sow Quick Ceck MATH MATH BOARD Te first prolem connects to te learning model. Have students use te MatBoard to explain teir tinking. Mat Talk Use Mat Talk to focus on students understanding of areas of triangles. MP5 Use appropriate tools strategically. Instruct students to draw an acute triangle and an otuse triangle on grid paper. Guide students to use a straigt-edge to determine te eigt of eac triangle. Use te cecked exercises for Quick Ceck. Students sould sow teir answers for te Quick Ceck on te MatBoard. 3 Rt I Sare and Sow. FInd te area of te triangle. A = _ A = _ 4 _ 8 A = _ 56 cm Find te area of te triangle. 5 in. 40 mm 7 mm On Your Own Area in. MATH BOARD 8 cm 4 cm Find te unknown measurement for te figure.. Te area of te triangle is 3 i n. Find te eigt of te triangle. = in. 4 mm 80 mm Area 5 7. cm = in. = 4.3 cm or 43 mm 6. in. A = 540 mm A = mm 5.5 mm Possile explanation: Te eigt of a triangle forms a 90º angle wit te ase. Mat Talk MATHEMATICAL PRACTICES 6 Explain ow you can identify te eigt of a triangle. If Ten a student misses te cecked exercises Differentiate Instruction wit Reteac 0.3 Personal Mat Trainer 6.G.A., 6.EE.A.c RtI Tier Activity (online) Hougton Mifflin Harcourt Pulising Company 7. MATHEMATICAL PRACTICE 3 Verify te Reasoning of Oters Te eigt of a triangle is twice te ase. Te area of te triangle is 65 i n. Carson says te ase of te triangle is at least 50 in. Is Carson s estimate reasonale? Explain. No. If te ase is 50 in., ten te eigt is 00 in. Using = 50 and = 00, A = _ (50)(00) =,500 in. Since,500 in. > 65 in., Carson's estimate of 50 in. is too ig. Capter 0 Lesson On Your Own Have students identify wat tey are looking for in eac exercise. For wic exercises will you find te eigt or ase? Exercises 5 and 6 MP3 Construct viale arguments and critique te reasoning of oters. How can you use te area and eigt of a triangle to determine te measure of te ase? Divide te area y te eigt. Multiply te quotient y to get te measure of te ase. PROBLEM TYPE SITUATIONS Multiplication and Division Area Unknown Product Exercise: 8 Area Unknown Factor Exercise: Capter 0

6 MATHEMATICAL PRACTICES ANALYZE Unlock te Prolem 8. DEEPER Alani is uilding a set of 4 selves. Eac self will ave supports in te sape of rigt isosceles triangles. Eac self is 4 inces deep. How many square inces of wood will se need to make all of te supports? a. Wat are te ase and eigt of eac triangle? Te ase and eigt are ot 4 in.. Wat formula can you use to find te area of a triangle? A = _ c. Explain ow you can find te area of one triangular support. Possile explanation: Multiply te ase and te eigt, and ten multiply te product y _ : _ 4 4 = _ 96 = 98. Te area of eac support is 98 in. 4 in. 4 in. 4 in. 4 in. d. How many triangular supports are needed to uild 4 selves? WRITE 8 supports Mat Sow Your Work e. How many square inces of wood will Alani need to make all te supports? 784 in. 4 ELABORATE Unlock te Prolem MATHEMATICAL PRACTICES MP3 Construct viale arguments and critique te reasoning of oters. Wen Alani first solved te prolem, se found tat se needed 98 square inces of wood. Wat mistake did Alani make? Se found te area of only one support, not te area of all 8 supports needed for four selves. Exercise 9 assesses te student s understanding of ow to use te formula for te area of a triangle to find one of te dimensions wen te area is known. Students can work ackward y multiplying te area y and ten dividing y Te area of a triangle is 97.5 cm. Te eigt of te triangle is 3 cm. Find te ase of te triangle. Explain your work cm; sustitute 97.5 for A and 3 for in te formula A = _ and solve for : 97.5 = _ 3; 97.5 = 6.5; 5 =. DIFFERENTIATED INSTRUCTION 0. Te area of a triangle is 30 ft. For numers 0a 0d, select Yes or No to tell if te dimensions given could e te eigt and ase of te triangle. 0a. = 3, = 0 Yes No 0. = 3, = 0 Yes No 0c. = 5, = Yes No 0d. = 5, = 4 Yes No INDEPENDENT ACTIVITIES Hougton Mifflin Harcourt Pulising Company Mat on te Spot Video Tutor Use tis video to elp students model and solve tis type of Tink Smarter prolem. Mat on te Spot videos are in te Interactive Student Edition and at Prolem 0 tests students aility to use a formula to find te area of a triangle. Tey are given te area and must determine wic dimensions could e te eigt and widt. If students select 0a, tey ave forgotten to multiply te ase and eigt y _. Encourage students to look for all possiilities, since tere are two possile correct pairs of dimensions. Differentiated Centers Kit 5 EVALUATE Formative Assessment Activities Complex Areas Students complete purple Activity Card 3 y using te formula for area and teir knowledge of polygons to find te area of complex polygons. Literature If I Designed te Zoo Students read tis ook and learn aout calculating te area and perimeter of complex sapes tat make up various animal enclosures in te zoo. Essential Question Using te Language Ojective Reflect Have students write instructions for tutoring a younger student to answer te essential question. How can you find te area of triangles? Use te formula A = _ to find te area of triangles. Mat Journal WRITE Mat Descrie ow you would find ow muc grass seed is needed to cover a triangular plot of land. Lesson

7 Practice and Homework Use te Practice and Homework pages to provide students wit more practice of te concepts and skills presented in tis lesson. Students master teir understanding as tey complete practice items and ten callenge teir critical tinking skills wit Prolem Solving. Use te Write Mat section to determine student s understanding of content for tis lesson. Encourage students to use teir Mat Journals to record teir answers. Area of Triangles Find te area.. A 5 _. 6 in. 5 in. A 5 _ A 5 45 Area 5 45 in. Find te unknown measurement for te triangle. 4. A = 0.5 mi = 0.6 mi = 0.75 mi 6. A = 63 m = = m 0.6 m. m 0.36 m 5. A = 4.86 yd = =.8 yd 5.4 yd 7. A =.5 km = 5 km = Practice and Homework Lesson 0.3 COMMON CORE STANDARD 6.G.A. Solve real-world and matematical prolems involving area, surface area, and volume ft ft 3 6 ft 0.5 m km Prolem Solving Hougton Mifflin Harcourt Pulising Company 8. Bayla draws a triangle wit a ase of 5 cm and a eigt of 8.5 cm. If se colors te space inside te triangle, wat area does se color? _ cm 0. WRITE Mat Descrie ow you would find ow muc grass seed is needed to cover a triangular plot of land. Ceck students work. 9. Alicia is making a triangular sign for te scool play. Te area of te sign is 558 in.. Te ase of te triangle is 36 in. Wat is te eigt of te triangle? _ 3 in. Capter COMMON ON CORE PROFESSIONAL DEVELOPMENT Mat Talk in Action Teacer: Rayanne: Teacer: Rayanne: Teacer: George: Teacer: Wat information do you know aout te triangle in Exercise 6? Te triangle as a eigt of meters and an area of 63 square meters. Wat are you asked to find? I need to find te ase of te triangle. How did you solve te prolem? I used te formula for te area of a triangle and I solved for, te ase. How did you solve for te ase? George: I multiplied _ times te eigt to get 6. Ten I divided te area y 6 to get 0.5 meters, te ase of te triangle. Teacer: Xavier: Teacer: Xavier: Does everyone tink tis is a good metod? Yes. Tat is ow I solved te prolem, too. How can you ceck your answer? I can sustitute te values for te eigt and te ase into te formula and evaluate to see if te result is te same as te area given in te prolem, 63 square meters. 549 Capter 0

8 Lesson Ceck (6.G.A., 6.EE.A.c). A triangular flag as an area of 87.5 square inces. Te ase of te flag measures 5 inces. How tall is te triangular flag?. A piece of stained glass in te sape of a rigt triangle as sides measuring 8 centimeters, 5 centimeters, and 7 centimeters. Wat is te area of te piece? Continue concepts and skills practice wit Lesson Ceck. Use Spiral Review to engage students in previously taugt concepts and to promote content retention. Common Core standards are correlated to eac section. 5 inces 60 square centimeters Spiral Review (6.EE.B.7, 6.EE.C.9, 6.G.A.) 3. Tina ougt a t-sirt and sandals. Te total cost was $4.50. Te t-sirt cost $8.95. Te equation c = 4.50 can e used to find te cost c in dollars of te sandals. How muc did te sandals cost? 4. Tere are 37 paper clips in a ox. Carmen places more paper clips in te ox. Write an equation to sow te total numer of paper clips p in te ox after Carmen places n more paper clips in te ox. $3.55 p = 37 + n 5. anoter ordered pair tat is on te grap of te equation represented y te tale. 6. Find te area of te triangle tat divides te parallelogram in alf. 550 People in group, x 3 4 Total cost of ordering lunc special ($), y Students answers will vary, ut te y-value of te ordered pair sould e 6 times as great as te x-value. Possile answers are (5, 30), (6, 36), and (7, 4). 3 cm 9 cm 58.5 square centimeters FOR MORE PRACTICE GO TO THE Personal Mat Trainer Hougton Mifflin Harcourt Pulising Company Lesson