GEOMETRY. Grades 6 8 Part 1 of 2. Lessons & Worksheets to Build Skills in Measuring Perimeter, Area, Surface Area, and Volume. Poster/Teaching Guide

Transcription

1 Grades 6 8 Part 1 of 2 Poster/Teacing Guide GEOMETRY Lessons & Workseets to Build Skills in Measuring Perimeter, Area, Surface Area, and Volume Aligned wit NCTM Standards Mat Grants Available

2 Lesson 1 Dear Teacer: Welcome to Setting te Stage wit Geometry, a new mat program aligned wit NCTM standards tat is designed to elp students in grades 6 8 build and reinforce basic geometry skills for measuring 2D and 3D sapes. Developed by Te Actuarial Foundation, tis program seeks to provide skill-building mat activities to elp your students become successful in te classroom and in real-world situations outside of scool. We ope you enjoy tis new program! Sincerely, Te Actuarial Foundation getting started In te lessons and workseets for tis program, students will learn and reinforce tese geometry skills: 1. measuring perimeter and area of 2D sapes; 2. measuring surface area of 3D sapes; and 3. measuring volume of 3D sapes. Te materials are taugt troug tis story line: A popular band called Te Geometrics is planning a big concert at a scool, but tey need elp to build a stage and promote te sow. Some students volunteer to become te Geometrics Stage Crew and use teir geometry knowledge to elp. Tree lesson plans teac basic measuring skills; eac lesson features a workseet, and is also supplemented by a bonus workseet and a take-ome activity. Before launcing te lessons, you can engage students in a discussion about real-world geometry wit te classroom poster. Sow your class ow geometric sapes can be found in te concert setting on te poster. Ask students were tey ave seen tese sapes in teir daily lives. Te poster includes fundamental formulas you can display in te classroom year-round. In addition, tere is a andy reference seet of formulas and definitions for teacers and students. Te reference seet also features drawings of all te sapes included in tese lessons. Note: All program pages appear in full color, yet are designed to easily reproduce in black and wite. Perimeter and Area of 2D Sapes Geometry Works! Te Stage Takes Sape OBJECTIVES: Students will understand formulas used to measure te perimeter and area of tese basic two-dimensional sapes: rectangles, circles, and triangles. Time Required: 20 minutes, plus additional time for workseets Materials: Student Workseet 1 Extensions: Bonus Workseet 1, Take-Home Activity 1 DIRECTIONS: 1. Review wit students te concept of perimeter. Perimeter is te total distance around te outside of a polygon (a closed figure made up of line segments). 2. On te board, draw a rectangle labeled wit a lengt of 4 feet and widt of 3 feet. Ten draw a rigt triangle wit a base of 4 feet, eigt of 3 feet, and ypotenuse (te side opposite te rigt angle) of 5 feet. Explain tat to measure te perimeter of any polygon, you add togeter te lengts of eac side. 3. Ask students wat te perimeter of te rectangle is. Sow students te formula for te rectangle s perimeter on te poster and ask wy it s correct. Te formula of P (perimeter) = 2 (l + w) is correct because a rectangle as two sets of sides tat are eac of equal lengt. Te perimeter of tis rectangle is 2 (4 + 3) = 14 feet. 4. Ask wat te perimeter of te triangle is. Sow tem te formula: P = side a + side b + side c. Te perimeter is , or 12 feet. 5. Tell students tat triangles can be classified by angles in tree ways: 1) rigt triangles wit one 90 angle were te base and eigt meet; 2) acute triangles wit all angles less tan 90 ; and 3) obtuse triangles wit one angle greater tan 90. Te angles of any triangle equal Draw a circle on te board. Draw a line from te center of te circle to te edge and mark it as 3 feet. Tell students tat tis is te radius. Ask tem wat te diameter is. (Te answer is 6 feet.) Ten explain tat te lengt of te line tat forms te circle is called te circumference. Tere is a unique formula for calculating te circumference: C (circumference) = p d (diameter). Tell students tat p is te circumference of any circle divided by its diameter and equals a number wit an infinite decimal: Te decimal continues on infinitely, but to solve most mat problems, people use a rounded ratio of Ask students to figure out te circumference of te circle you ave drawn. Ask tem to provide te answer to te nearest alf foot. As = feet, te answer is 19 feet. 7. Now go over te definition of area on te poster: te measure of a bounded region of a two-dimensional sape expressed in square units, e.g., square inces or square feet. Sow your students te formula for area of a rectangle: A (area) = l w. Ask tem to calculate te area of te rectangle you ad drawn earlier (4 3 = 12 square feet). 8. Now point out te formula for te area of a triangle on te poster: A = 1/2 [b (base) (eigt)]. Refer back to your drawing of a rigt triangle wit a base of 4 feet and eigt of 3 feet. Ask students to calculate te area. Te answer is 1/2 (4 3) = 6 square feet. 9. Finally, go over te area formula for circles. Again, refer to te poster: A = p r 2, were r 2 means radius squared, or r r. Te answer is p (3.14) r 2 (3 3) = square feet. Ask students to provide te answer to te nearest alf foot. Te answer is 28.5 feet or 28 feet and 6 inces. 10. Distribute Student Workseet 1. Tell students tey sould complete all te questions. Explain tat te bonus question introduces a new formula for te area of trapezoids. Go over correct answers as a class using te Workseet Answer Key (see back cover).

3 Lesson 2 Surface Area of 3D Sapes Tat Sould Cover It! OBJECTIVES: Students will understand formulas used to measure te surface area of tese basic tree-dimensional sapes: a rectangular prism, a cylinder, and a square pyramid. Time Required: 20 minutes, plus additional time for workseets Materials: Student Workseet 2 Extensions: Bonus Workseet 2, Take-Home Activity 2 DIRECTIONS: 1. After mastering te area of 2D sapes, students can now learn te formulas to measure 3D sapes. 2. Draw a rectangular prism on te board wit tese measurements: eigt = 3 feet, lengt = 4 feet, and widt = 5 feet. 3. Sow students te surface area formula for rectangular prisms on te poster: SA = 2 (l w + l + w ). Explain to tem tat te surface area of 3D objects is measured in square units, just like te area of 2D objects, and is te sum of all of te 3D object s surfaces. 4. Ask students wat te surface area is of te sape you ave drawn. Te answer is 2 ( ) = 94 square feet. 5. Now draw a cylinder and mark te dimensions wit te radius at 3 feet and te eigt at 4 feet. 6. Sow students te surface area formula for cylinders on te poster: SA = (2 p r 2 ) + (p d ) and ask tem to calculate te answer to te nearest alf foot. As ( ) + ( ) = square feet, te answer is 132 square feet. 7. Finally, draw a square pyramid on te board and mark te dimensions wit a base lengt of 6 feet and a base widt of 6 feet. Sow te slant eigt as 5 feet by drawing a perpendicular line from te center of one of te base sides to te top of te pyramid. Te square pyramid as a base area (BA) measurable by l w like any square or rectangle. 8. Sow students te surface area formula for square pyramids on te poster, SA = (BA) + 1/2 P slant and ask students to calculate te answer. Tis formula adds togeter te area of te base wit te area of te four triangular sides of te square pyramid. Te P in te formula refers to te perimeter of te base. Te answer is / = 96 square feet. 9. Distribute Student Workseet 2. Tell students tey sould complete all te questions. You may want to take some extra time in class to go over te bonus question, wic introduces te formula for measuring te surface area of a cone [SA = (p r 2 ) + (p r slant)]. Go over all correct answers as a class, referring to te Workseet Answer Key (see back cover). Lesson 3 Volume of 3D Sapes Pack It Up! Wat Will Fit? OBJECTIVES: Students will understand formulas used to measure te volume of tese basic tree-dimensional sapes: a rectangular prism, a cylinder, and a square pyramid. Time Required: 20 minutes, plus additional time for workseets Materials: Student Workseet 3 Extensions: Bonus Workseet 3, Take-Home Activity 3 DIRECTIONS: 1. Explain to your students tat now tat tey ve mastered measuring te surface area of 3D sapes, tey can move on to measuring volume, wic is te amount of space inside a 3D sape, measured in cubic units. Refer to te poster, wic provides essential formulas. 2. Again, draw a rectangular prism on te board like te one from te Lesson 2 surface area unit wit tese measurements: eigt = 3 feet, lengt = 4 feet, and widt = 5 feet. 3. Sow students te volume formula for rectangular prisms on te poster: V (volume) = l w. Ask students wat te volume of te prism is. Te answer is = 60 cubic feet. 4. Now draw a cylinder again wit te same dimensions as in Lesson 2: Te radius is 3 feet and te eigt is 4 feet. 5. Sow students te volume formula for cylinders on te poster: V = p r 2. Ask students wat te volume of te cylinder is wen rounded to te nearest alf foot. As = cubic feet, te answer is 113 cubic feet. 6. You migt add tat a cylinder is like a barrel, and volume measurement can elp determine ow muc liquid will fit in a container tis size. One cubic foot = 7.48 gallons. Ask students ow muc water tis cylinder would old. Te answer is , or gallons (wen rounded to te nearest undredt). Students may need a calculator to solve tis problem. 7. Finally, draw a square pyramid on te board wit te same dimensions as in Lesson 2: Te square pyramid as a base lengt of 6 feet and a base widt of 6 feet. Te eigt of te pyramid is 4 feet. 8. Sow students te volume formula for square pyramids on te poster: V = 1/3 BA. Ask students for te volume of te square pyramid. Te answer is 1/ = 48 cubic feet. 9. Distribute Student Workseet 3. Tell students tey sould complete all te questions. You may want to take some extra time to go over te bonus question, wic introduces te formula for te volume of a cone [V = p 1/3 r 2 ]. Go over all correct answers as a class referring to te Workseet Answer Key (see back cover). Look for more mat resources at Te Actuarial Foundation Web site at: Advancing Student Acievement Grants Expect te Unexpected Wit Mat Series: Sake, Rattle, & Roll (probability) Bars, Lines, & Pies (graping) Conversions Rock (converting decimals, fractions, and percents) Te Mat Academy Series: Using Mat in te Real World Also look for printable program copies at:

4 Reference Seet Perimeter, Area, Surface Area, and Volume: Review of Terminology, Basic Sapes, and Formulas area: te measure of a bounded region of a two-dimensional sape expressed in square units circumference: te distance around te edge of a circle diameter: te distance across a circle troug its center point ypotenuse: te side opposite te 90 angle in a rigt triangle, also te longest side of a rigt triangle perimeter: te total distance around te outside of a polygon pi or p: te circumference of any circle divided by its diameter, rounded to te number 3.14 radius: te measure from te center of a circle to a point on te circle terminology slant: te diagonal distance from te top of a cone to its base slant eigt: te eigt of one of te triangular faces of a pyramid surface area: te sum of all te areas of all surfaces of a tree-dimensional object, measured in square units volume: te amount of space inside a tree-dimensional sape, measured in cubic units abbreviations: d = diameter r = radius A = area = eigt SA = surface area b = base l = lengt slant = slant eigt BA = base area P = perimeter V = volume C = circumference p = pi = 3.14 w = widt basic sapes and formulas 2D Sapes: Perimeter and Area 3D Sapes: Surface Area and Volume Rectangle P = 2 (l + w) A = l w w l Rectangular Prism SA = 2 (l w + l + w ) V = l w l w Triangle P = side a + side b + side c A = 1/2 (b ) ypotenuse b b b obtuse Triangle acute Triangle Rigt Triangle Square Pyramid SA = (BA) + 1/2 P slant V = 1/3 BA Note: base area (BA) of a square or rectangular pyramid is l w of te base, and P is perimeter of te base. l slant w Circle C = p d A = p r 2 d r Cylinder SA = (2 p r 2 ) + (p d ) V = p r 2 r Trapezoid P = side a + b1 + b2 + side c A = 1/2 (b1 + b2) a b1 b2 c Cone SA = (p r 2 ) + (p r slant) V = p 1/3 r 2 slant r

5 Student Workseet 1 Name: Date: Geometry Works! Te Stage Takes Sape Te popular band Te Geometrics wants to play a special concert at your scool, but tey need a stage crew to elp. Te first step for te Geometrics Stage Crew is building an elaborate stage featuring differently saped sections. 1 First, tey want a main stage tat is rectangular-saped, measuring a lengt of 24 feet and a widt of 16 feet. Wat are te perimeter and area of tat stage? Perimeter: Area: 2 Second, te band s lead guitarist wants te Geometrics Stage Crew to build a smaller circular stage in front of te main stage tat e can step onto and play a solo. Te diameter as to be one-tird of te lengt of te main stage. Wat is te circumference and area? Round your answer to te nearest foot. Circumference: Area: 3 Te bass player as a ting for triangles and sees erself on a triangular platform off to te left of te stage. Wen viewed from above, te rigt triangle as a eigt of 8 feet, a base of 6 feet, and a tird side (called te ypotenuse) of 10 feet. Wat is te perimeter and area? Perimeter: Area: Te drummer wants to be on a raised trapezoid-saped platform. Bonus: Tis requires te Geometrics Stage Crew to learn a new formula for te area of trapezoids [A = 1/2 (b1 + b2) ]. Te trapezoid is sown in te diagram ere wit te measurements indicated. Base 1 (b1) = 8 feet. Base 2 (b2) = 5 feet. Te eigt measures 6 feet. Wat is te area? 5 ' 6 ' 8 '

6 Student Workseet 2 Name: Date: Tat Sould Cover It! Te Geometrics love sapes. For te upcoming concert, te tree main players eac want to emerge from uman-size sapes of a rectangular prism, a cylinder, and a square pyramid. Wile tey already ave tese props built, te band asks te Geometrics Stage Crew to paint over tem completely (even te bottom of eac object). Te stage crew knows tat 1 gallon of paint covers 350 square feet. To buy te rigt amount of paint, te stage crew as to calculate te surface area of eac sape. 1 Te dimensions of te rectangular prism for te lead guitarist are eigt = 7 feet, widt = 4 feet, and lengt = 3 feet. Surface Area = 3 Te dimensions of te square pyramid for te bass player are base lengt = 4, base widt = 4, and a slant eigt of 7.28 feet. Solve wit a decimal, ten also round to te nearest alf foot. Surface Area = 2 Te dimensions of te cylinder for te drummer are radius = 3.5 feet and eigt = 7 feet. Solve wit a decimal, ten also round to te nearest alf foot. Surface Area = 4 Can te stage crew paint te surface area of all tree sapes wit just one can of paint? Te drummer says tat e d also like to ave a backup cone wit te Bonus: same eigt and radius as te cylinder and a slant of 7.8 feet. Te stage crew needs a new formula to figure out te surface area of a cone: SA = (p r 2 ) + (p r slant). 7' 7.8' a. Wat is te surface area? Express as a decimal and ten round to te nearest foot. 3.5' b. if one gallon of paint covers 350 square feet, about ow muc of a gallon is needed to paint te cone?_

7 Student Workseet 3 Name: Date: Pack It Up! Wat Will Fit? Te Geometrics Stage Crew now as to transport te painted props of a rectangular prism, a cylinder, and a square pyramid to te concert. Tey ave to make sure te van tey ave is big enoug to carry te props. To do tis, tey are going to measure te volume of te cargo space and compare tat to te volume of te tree objects tey ave. Te dimensions of te objects again are: Rectangular Prism: eigt = 7 feet, widt = 4 feet, and lengt = 3 feet Cylinder: radius = 3.5 feet and eigt = 7 feet Square Pyramid: base lengt = 4, base widt = 4, a slant eigt of 7.28 feet, and a eigt of 7 feet 1 First, get te complete volume of all te objects combined. a. Wat is te volume of te rectangular prism? b. Wat is te volume of te cylinder? (Give te decimal answer and ten round it to te nearest cubic foot.) c. Wat is te volume of te square pyramid? d. Wat is te total volume of all te objects combined, rounded to te nearest foot? 2 Te van cargo space measures 8 feet tall by 5 feet wide by 13 feet deep. Wat is te volume of te cargo space? 3 Based on te volume measurements, can you estimate if te objects will fit in te van? based on volume, would tere still be room for te cone from te Bonus: bonus section of Workseet 2? Te radius of te cone = 3.5 feet and eigt = 7 feet and slant = 7.8 feet. To calculate te volume (cubic measurement) of te cone, you need to learn a new formula: V = p 1/3 r 2. 7' 7.8' 3.5'

8 workseet answer key Student Workseet 1: Geometry Works! Te Stage Takes Sape 1. Perimeter: 2 ( ) = 80 feet Area: = 384 square feet 2. 8 feet is one-tird te lengt of te main stage. Circumference: = feet, rounded = 25 feet Area: = square feet, rounded = 50 square feet 3. Perimeter: = 24 feet Area: 1/2 (8 6) = 24 square feet Bonus: Trapezoid Area = 1/2 (8 + 5) 6 = 39 square feet Student Workseet 2: Tat Sould Cover It! 1. 2 ( ) = 2 ( ) = 2 61 = 122 square feet 2. ( ) + ( ) = (76.93) + (153.86) = square feet, rounded = 231 square feet 3. (4 4) + 1/ = = square feet, rounded = 74 square feet 4. No. Te total surface area is square feet and one can of paint will cover only 350 square feet, so te stage crew needs more paint. Bonus: a. ( ) + ( ) = = square feet, rounded = 124 square feet; b /350 = or about.35, or 35% of te gallon Student Workseet 3: Pack It Up! Wat Will Fit? 1. a = 84 cubic feet b = cubic feet, rounded = 269 cubic feet c. 1/3 (4 4) 7 = cubic feet, rounded = 37cubic feet d = 390 cubic feet = 520 cubic feet 3. Yes, because te total volume of te objects is 390 cubic feet, leaving extra room volume-wise. Bonus: Te cone as a volume of / = cubic feet or 89 cubic feet and 1,296 cubic inces (1,728 cubic inces = 1 cubic foot). Tere was approximately 130 cubic feet left in te van. So based on volume alone, tere sould still be enoug room in te van to fit te cone. Bonus Workseet 1: Wat s te Angle? For a wood floor, te wall angles are: [180 ( )] = 10 for a maximum safe floor angle, and [180 ( )] = 22 for a minimum safe floor angle. 1. Since 22 is LARGER tan 10, ten 22 is te MAXIMUM safe wall angle. 2. Since 10 is SMALLER tan 22, ten 10 is te MINIMUM safe wall angle. For a carpet, te wall angles are: [180 ( )] = 5 for a maximum safe floor angle, and [180 ( )] = 60 for a minimum safe floor angle. 3. Since 60 is LARGER tan 5, ten 60 is te MAXIMUM safe wall angle. 4. Since 5 is SMALLER tan 60, ten 5 is te MINIMUM safe wall angle. 5. 1/2 (8 14) = 56 square inces Bonus Workseet 2: Tat s a Wrap! 1. Te lengt of te poster is te same as te tras can s eigt and te widt of te poster is equal to te tras can s circumference. Te circumference is = 9.42 feet. Te poster s dimensions are 4 feet long and 9.42 feet wide, or 4 feet by 9.5 feet rounded to te nearest alf foot. 2. Te surface of te tras cans witout te top and bottom can be derived using part of te formula for a cylinder s surface area: SA = p d = square feet or 38 square feet wen rounded to te nearest square foot. 3. Using te surface area formula for a rectangular prism, eac CD as a surface area of: 2 ( ) = 65.5 square inces. Multiply te surface area of 1 CD 100 to find te total amount of paper needed to wrap 100 CDs: = 6,550 square inces of paper. 4. Te answer uses te surface area formula for a square pyramid but witout te base area: 1/2 40 feet 10 feet = 200 square feet. Bonus Workseet 3: Turn Up te Volume! 1. Te volume of te room is 180,000 cubic feet, so te band can turn up teir amplifiers 10 notces in tis gym. 2. First calculate te area of te gym floor: = 6,000 square feet. If 1,200 people fit into 6,000 square feet, ten one person occupies 5 square feet (6,000 1,200 = 5). For 1,500 people: 1,500 5 = 7,500 square feet. 3. In te formula for te volume of a rectangular prism (V = l w ), te l w is actually te area of te floor, so you can say V = floor area eigt. Rearrange te formula to: Heigt = Volume floor area. H = 280,000 8,000 = 35 feet. 4. Using te formulas for te volume of a square pyramid and te area of a rectangle, students can find te lengt of one of te ologram s base sides: 6,250 = 1/3 BA 30, so BA = 625. Because BA = l w and = 625, one side of te pyramid s base is 25 feet. Take-Home Workseet Front Cover: Warm-Up 1. area; 2. surface area; 3. volume; 4. cylinder; 5. cone Take-Home Activity 1: Poster-Crazy 1. Answers will vary 2. Area of rectangular posters: = = 935 square inces, or 6.5 square feet Area of circular posters: p 1 2 = 3.14 square feet 5 = 15.7 square feet Area of triangular poster: 1/2 (3 3) = 4.5 square feet Total area of posters: 26.7 square feet Now Try Tis: Answers will vary depending on te widt of te doorway, wic will determine te diameter of te welcome mats. Students need to measure te diameter and put teir numbers in te formulas for circumference and area. Take-Home Activity 2: Covering Up 1. Te student is painting 2 sides and a top eac measuring 3 square feet, and te back measuring 9 square feet = 18 square feet 2. (75 54) + [2(75 6)] + [2(54 6)] = 4,050 square inces square inces square inces = 5,598 square inces. 144 square inces = 1 square foot, so 5,598/144 = square feet, or 38 square feet and 126 square inces. Now Try Tis: To figure out te area of te at, use te formula p r slant = square inces. If a 1-ounce jar covers 33 square inces, te student does not ave enoug paint for is or er at. Take-Home Activity 3: Te Perfect Fit = 7.5 cubic inces; = 600 cubic inces 2. Answers will vary = 3,532.5 cubic feet, or 26,423.1 gallons Now Try Tis: V = p r 2, so 22 = To get te answer for, divide 22 by = Rounded to te nearest inc te eigt is 7 inces. alignment wit standards: National Council of Teacers of Matematics (NCTM) Tis program aligns wit NCTM Geometry Standards for Grades 6 8: ttp://standards.nctm.org/document/capter6/geom.tm