Unit 5 Area, the Pythagrean Therem, and Vlume Unit Gals Stage 1 Gemetry ACC Number f : 34 2/27/17 4/13/17 Unit Descriptin: Deriving new frmulas frm previusly discvered nes, the students will leave Unit 5 with an understanding f area and vlume frmulas fr plygns, circles and their three-dimensinal cunterparts. Unit 5 seeks t have the students develp and describe a prcess whereby they are able t slve fr areas and vlumes in bth prcedural and applied cntexts. Units f measurement are emphasized as a methd f checking ne s apprach t a slutin. Using their study f circles frm Unit 3, the students will dissect the Pythagrean Therem. Thrugh this they will develp prperties f the special right triangles: 45, 45, 90 and 30, 60, 90. Algebra skills frm previus classes lead students t understand the cnnectin between the distance frmula and the Pythagrean Therem. In all instances, surface area will be generated using the smaller pieces frm which a given shape is cnstructed. The student will, again, derive new understandings frm ld nes. Materials: Cnstructin tls, cnstructin paper, patty paper, calculatrs, Desms, rulers, glue sticks (ptinal), sets f gemetric slids, pennies and dimes (Explre p. 458), string, prtractrs, paper clips, square, ismetric and graph paper, sand r water, plastic dishpan Standards fr Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense f prblems and persevere in slving them. Reasn abstractly and quantitatively. Cnstruct viable arguments and critique the reasning f thers. Mdel with mathematics. Use apprpriate tls strategically. Attend t precisin. Lk fr and make use f structure. Lk fr and express regularity in repeated reasning. Standards fr Mathematical Cntent Clusters Addressed [m] G-SRT.C Define trignmetric ratis and slve prblems invlving right triangles. [a] G-C.B Find arc lengths and areas f sectrs f circles. Transfer Gals Students will be able t independently use their learning t Make sense f never-befre-seen prblems and persevere in slving them. Cnstruct viable arguments and critique the reasning f thers. Making Meaning UNDERSTANDINGS Students will understand that By cutting apart and rearranging figures, new frmulas can be derived frm ld nes. Units are a key t differentiating between area, u 2, and vlume, u 3. The Pythagrean Therem leads t the distance frmula and the equatin f a circle. In a right triangle, the area f the square n the hyptenuse is the sum f the areas f the squares n the legs. Pi is the cnstant f prprtinality between the circumference and the diameter f a circle. If a shape is divided int subregins, the sum f the areas f thse subregins, even if they are rearranged, is equal t the area f the riginal shape. When a triangle r parallelgram is rtated s that a different side is its base, the area f that triangle r parallelgram hlds cnstant. T find the surface area f a prism, pyramid, cylinder r cne, cmbine the shape s lateral surface area with the area(s) f the shape s base(s). Sub-shapes r near-shapes can be used t apprximate surface area and vlume. ESSENTIAL QUESTIONS Students will keep cnsidering D we need t memrize new frmulas r can we use ld frmulas in a new way? If we have a shape with a given area, and we cut that shape apart and rearrange its pieces, will the area remain cnstant? Hw are the area frmulas fr a square, a rectangle, a trapezid, and a triangle related? Why d the slutin pints fr 2 2 2 ( x h) + ( y k) = r frm a circle? Hw can yu find the vlume f a slid fr which n frmula is available? Hw can a cncept map r a Venn diagram be cnstructed t rganize the slids based n their cmmnalities? Hw is finding the vlume f a prism similar t finding the area f a rectangle? 2016-2017 1 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume [a] G-GPE.A Translate between the gemetric descriptin and the equatin fr a cnic sectin. [m] G-GPE.B Use crdinates t prve simple gemetric therems algebraically. [a] G-GMD.A Explain vlume frmulas and use them t slve prblems. [m] G-MG.A Apply gemetric cncepts in mdeling situatins. Unit Gals Stage 1 Acquisitin KNOWLEDGE Students will knw Frmulas and methds fr finding the areas f rectangles, parallelgrams, triangles, trapezids, kites, regular plygns, circles, sectrs, segments, and annuluses. The Pythagrean Therem and that, given a 2 + b 2 = c 2, a and b are the lengths f the legs. The length f the hyptenuse is c. The Cnverse f the Pythagrean Therem. If tw figures are similar with a scale factr s, then their areas have scale factr s 2. The area f a sectr is the same part f the area f the whle circle as the measure f the central angle f the sectr is f 360. In a 45-45 -90 triangle, if the legs have length m, then the hyptenuse has length m 2. In a 30-60 -90 triangle, if the leg ppsite the 30 has length a, then the hyptenuse has length 2a and the ther leg as length a 3. 2 2 2 The equatin ( x h) + ( y k) = r describes a circle with pints n the circle (x, y), a radial distance f r, and center (h, k). The vlume frmulas fr prisms, cylinders, pyramids, cnes, and spheres. The surface area frmula fr a sphere. Gemetry ACC SKILLS Students will be skilled at and/r be able t Derive frmulas fr the areas f rectangles, parallelgrams, triangles, trapezids, kites, regular plygns, circles, annuluses, sectrs, segments f circles, and surface areas f spheres. Derive the fact that the length f the arc intercepted by an angle is prprtinal t the radius. Apply surface area frmulas t slve prblems. Slve area applicatin prblems using varius prblem-slving strategies. Slve fr the missing side length f a triangle using the Pythagrean Therem. Use the Cnverse f the Pythagrean Therem t prve that a triangle is right. Slve fr the lengths f the sides f a 45-45 -90 triangle and a 30-60 -90 triangle. Derive the distance frmula frm the Pythagrean Therem. Use the distance frmula t slve prblems. Derive the equatin f a circle f given center and radius using the Pythagrean Therem. Apply the Pythagrean relatinship t prblems invlving circles. Use crdinates t cmpute perimeters f plygns and areas f triangles and rectangles. Derive frmulas fr finding the vlumes f prisms, cylinders, pyramids, cnes, and spheres using dissectin arguments and Cavalieri s Principle. Slve applied prblems invlving plyhedra, cnes, cylinders, spheres, r hemispheres. Use displacement t find the vlumes f irregularly shaped slids. 2016-2017 2 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Assessed Grade Level Standards Gemetry ACC Standards fr Mathematical Practice SMP 1 Make sense f prblems and persevere in slving them. SMP 2 Reasn abstractly and quantitatively. SMP 3 Cnstruct viable arguments and critique the reasning f thers. SMP 4 Mdel with mathematics. SMP 5 Use apprpriate tls strategically. SMP 6 Attend t precisin. SMP 7 Lk fr and make use f structure. SMP 8 Lk fr and express regularity in repeated reasning. Standards fr Mathematical Cntent [m] G-SRT.C Define trignmetric ratis and slve prblems invlving right triangles. G-SRT.6 Understand that by similarity, side ratis in right triangles are prperties f the angles in the triangle, leading t definitins f trignmetric ratis fr acute angles. [a] G-C.B Find arc lengths and areas f sectrs f circles. G-C.5 Derive using similarity the fact that the length f the arc intercepted by an angle is prprtinal t the radius, and define the radian measure f the angle as the cnstant f prprtinality; derive the frmula fr the area f a sectr. [a] G-GPE.A Translate between the gemetric descriptin and the equatin fr a cnic sectin. G-GPE.1 Derive the equatin f a circle f given center and radius using the Pythagrean Therem; cmplete the square t find the center and radius f a circle given by an equatin. [m] G-GPE.B Use crdinates t prve simple gemetric therems algebraically. G-GPE.7 Use crdinates t cmpute perimeters f plygns and areas f triangles and rectangles, e.g., using the distance frmula. [a] G-GMD.A Explain vlume frmulas and use them t slve prblems. G-GMD.1 Give an infrmal argument fr the frmulas fr the circumference f a circle, area f a circle, vlume f a cylinder, pyramid, and cne. Use dissectin arguments, Cavalieri s Principle, and infrmal limit arguments. G-GMD.2 (+) Give an infrmal argument using Cavalieri s principle fr the frmulas fr the vlume f a sphere and ther slid figures. G-GMD.3 Use vlume frmulas fr cylinders, pyramids, cnes, and spheres t slve prblems. [m] G-MG.A Apply gemetric cncepts in mdeling situatins. G-MG.1 Use gemetric shapes, their measures, and their prperties t describe bjects (e.g., mdeling a tree trunk r a human trs as a cylinder). G-MG.2 Apply cncepts f density based n area and vlume in mdeling situatins (e.g., persns per square mile, BTUs per cubic ft). Key: [m] = majr clusters; [s] = supprting clusters, [a] = additinal clusters Indicates a mdeling standard linking mathematics t everyday life, wrk, and decisin-making CA Indicates a Califrnia-nly standard 2016-2017 3 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Assessment Evidence Evidence f Stage 2 Gemetry ACC Unit Assessment Claim 1: Students can explain and apply mathematical cncepts and carry ut mathematical prcedures with precisin and fluency. Cncepts and skills that may be assessed in Claim 1: [m] G-SRT.C Students will use similarity t shw that side ratis in right triangles are prperties f the angles in the triangle. [a] G-C.B Students will, using similarity, derive the fact that the length f the arc intercepted by an angle is prprtinal t the radius. Students will derive the frmula fr the area f a sectr. [a] G-GPE.A Students will use the Pythagrean Therem t derive the equatin f a circle when given its center and radius. [m] G-GPE.B Students will use crdinates t cmpute the perimeters f plygns. Students will use crdinates t cmpute the area f triangles and rectangles. [a] G-GMD.A Students will give an infrmal argument fr the area f a circle. Students will give an infrmal argument fr the vlume f a cylinder, pyramid and cne. Students will use dissectin arguments and Cavalieri s Principle. Students will use frmulas t slve vlume prblems invlving cylinders, pyramids, cnes and spheres. [m] G-MG.A Students will use gemetric shapes, their measures, and their prperties t describe bjects. Students will apply cncepts f density based n area and vlume in mdeling situatins. Claim 2: Students can slve a range f wellpsed prblems in pure and applied mathematics, making prductive use f knwledge and prblem-slving strategies. Standard clusters that may be assessed in Claim 2: G-SRT.C Claim 3: The student can clearly and precisely cnstruct viable arguments t supprt their wn reasning and critique the reasning f thers. Standard clusters that may be assessed in Claim 3: Nne Claim 4: The student can analyze cmplex, real-wrld scenaris and can cnstruct and use mathematical mdels t interpret and slve prblems. Standard clusters that may be assessed in Claim 4: G-MGD.A G-MG.A 2016-2017 4 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Other Evidence Evidence f Stage 2 Gemetry ACC Frmative Assessment Opprtunities Opening Tasks Infrmal teacher bservatins Checking fr understanding using active participatin strategies Exit slips/summaries Mdeling Lessns (SMP 4) Tasks Frmative Assessment Lessns (FAL) Quizzes/Chapter Tests SBAC Interim Assessment Blcks Access Using Frmative Assessment fr Differentiatin fr suggestins. Lcated n the LBUSD website M Mathematics Curriculum Dcuments 2016-2017 5 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume 1-2 I will explre the relatinship between the side length and vlume f a rectangular bx. Suggested Sequence f Key Events and Instructin Discvering Gemetry Expectatins Lessns) Opening Task - The Open-Tpped Bx T explre the relatinship between length and vlume, students design and cnstruct an pen-tpped bx. Once the bx is cmplete, they slve fr surface area and vlume. After gathering the data nt a scatter plt (x = crner edge length; y = vlume), students use a graphing prgram t slve fr a regressin line. Using that line, students can discuss data trends, dmain and range, and decide which f the bxes has the greatest vlume based n the height f the bx. The Supplemental Resurces n the prvide vides f students in the prcess f a similar activity, an article with teacher suggestins fr cnducting this activity, and instructins n creating scatterplts and regressin lines with the TI-84. This activity serves t mtivate and review tpics that will be used thrughut Unit 5. Gemetry ACC Applicatin: The Open- Tpped Bx 4-5 I will investigate the area f plygns by Deriving frmulas fr areas f rectangles, parallelgrams, triangles, trapezids and regular plygns. Rearranging figures t derive new frmulas frm previusly discvered frmulas. Slving area prblems which require several steps. Describing the prcess used when slving area prblems. Answering questins such as What variables have yu seen used in area frmulas fr rectangles? What d thse variables represent? Hw d yu knw which side is the length and which side is the width? Hw are rectangles and parallelgrams the same? Different? Hw can the area frmula fr a rectangle be the same as the area frmula fr a parallelgram? Is there mre than ne way that a parallelgram can be cut up and rearranged int a rectangle? Lessn 8-1 Lessn 8-2 Lessn 8-3 Lessn 8-4 Cnceptual Illuminatins: Area Frmulas 2016-2017 6 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Suggested Sequence f Key Events and Instructin Discvering Gemetry Expectatins Lessns) D all rectangles with the same perimeter have the same area? Hw are all these area frmulas similar? Different? Can the triangle area frmula be used t find area frmulas fr trapezids and kites? Hw d yu knw what side is the base f a triangle? What happens t the frmula fr the area f a trapezid if ne f the bases shrinks t length zer? When slving fr area, in what cntext wuld yur measurements need exact precisin? Using the triangle area frmula, hw can yu find the area f any regular plygn? Why d all the issceles triangles frmed within a regular n-gn have the same area? Hw clse is the area f an n-gn t the area f the circumscribed circle? Hw culd yu increase that clseness? Gemetry ACC 3-4 I will slve prblems invlving circles by Deriving new frmulas frm ld t slve fr the area f a: circle, sectr, annulus, and segment. Applying new frmulas t slve multi-step prblems. Applying area frmulas t slve fr gemetric prbabilities. Finding the surface areas f slids including cnes and cylinders. Answering questins such as 2 What are ways t remember that area is π r and circumference is 2π r? Why is π in these frmulas? Hw lng d we carry π alng in ur slutins rather than runding t decimals? Why? When wuld we want t express ur answers using the symbl π and when wuld we want t use a decimal equivalent? Given a slice f pizza, hw can yu determine what part the slice is f the entire area f the pizza? Lessn 8-5 Lessn 8-6 Lessn 8-7 Explre p. 458 Cnceptual Open Middle: Which Circle Is Bigger? Applicatin: Illuminatins: Gerrymandering Is It r Isn t It 3-Act Lessn: Rtunda West Flrida 3-Act Lessn: Hly Cw MathVisin Prject: Pied 2016-2017 7 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Suggested Sequence f Key Events and Instructin Discvering Gemetry Expectatins Lessns) What is the difference between a sectr, a segment, an annulus and an arc? Describe hw yu find the segment f a circle. a 2 a In the frmula, A = πr, what des the represent? What 360 360 is its purpse? D yu need t memrize these new frmulas? What shapes can yu break rectangular prisms, pyramids, cnes, and cylinders int? Hw d yu slve fr the area f each f these individual shapes? What are sme f the differences between surface area and vlume? Is there a relatinship between the surface area frmulas fr cylinders, pyramids and cnes? Hw wuld yu derive the simplest frmulas fr the ttal surface area f a prism, square-based pyramid, cylinder and cne? Gemetry ACC MathVisin Prject: Madisn s Rund Garden Illustrative Mathematics: Setting Up Sprinklers 2-3 I will investigate the Pythagrean Therem and its cnverse by Dissecting the squares n the legs f a right triangle and fitting the pieces int the square n the hyptenuse t prve the Pythagrean Therem. Applying the Pythagrean Therem t find missing lengths in right triangles. Using the Cnverse f the Pythagrean Therem t determine right triangles. Answering questins such as Why is the lngest leg always ppsite the right angle in a right triangle? What is the difference between a therem and a cnjecture? Des the Pythagrean Therem wrk fr ALL triangles? 2 2 2 If the sides f a triangle satisfy a + b = c, what can yu say abut the triangle? Remember, the cnverse f a true statement is nt always true. Lessn 9-1 Lessn 9-2 Cnceptual OpenMathRef: A Graphical Prf f the Pythagrean Therem 2016-2017 8 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Suggested Sequence f Key Events and Instructin Expectatins If a triangle has side lengths that are a Pythagrean triple and anther triangle has side lengths that are a multiple f that Pythagrean triple, what s the relatinship between thse tw triangles? Discvering Gemetry Lessns) Gemetry ACC 2-3 I will apply the Pythagrean Therem and its cnverse by Explring the special prperties f special right triangles: 45-45 -90 and 30-60 -90. Slving multi-step real-wrld prblems. Answering questins such as What d yu always knw abut an issceles right triangle? What d yu always knw abut a right triangle with a leg that is half the length f the hyptenuse? Knwing the side length ratis f the special right triangles will be necessary in math class up thrugh Precalculus and Calculus. What can yu d t help yu remember these side length ratis? What are steps that yu can take t slve a wrd prblem? Lessn 9-3 Lessn 9-4 Prcedural Skills and Fluency: Gegebra Interactive: Special Right Triangles 1-2 I will use crdinates in gemetry by Using crdinates t derive the distance frmula. Recgnizing the cnnectin between the distance frmula and the Pythagrean Therem. Using the center and a pint n the circumference t slve fr the equatin f a circle. Slving prblems using the distance frmula. Explring rates f change via the Pythagrean Therem. Answering questins such as Hw wuld yu use a right triangle t find the distance between tw pints? If yu are ging t use a calculatr t slve a distance prblem, what are sme things that yu can d t be mre likely t get a crrect answer? Hw imprtant is it t be exact with yur substitutin int the distance frmula s x, ) and x, )? ( 1 y1 ( 2 y 2 Lessn 9-5 Explre p. 507 Cnceptual Illustrative Mathematics: Triangle Perimeters Prcedural Skills and Fluency: MathOpenRef: Trapezid (Crdinate Gemetry) 2016-2017 9 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume 1-2 I will use the Pythagrean Therem in circle gemetry by Suggested Sequence f Key Events and Instructin Discvering Gemetry Expectatins Lessns) Slving multi-step prblems. Lessn 9-6 Cmbining different cnjectures t slve a variety f prblems. Answering questins such as Hw did yu cmbine cnjectures, frmulas and prperties t slve prblems in this sectin? Gemetry ACC 4-5 I will investigate the vlumes f slids by Identifying prisms, pyramids, cnes and cylinders and describing their attributes. Investigating relatinships between the vlumes f prisms and pyramids, and cylinders and cnes. Visualizing three-dimensinal bjects using mdels and nets. Cnstructing infrmal arguments fr the vlumes f prisms, pyramids, cylinders, and cnes. Answering questins such as What is the difference between a plyhedrn and a plygn? Hw d yu knw hw many faces a plyhedrn has? Where are the base, vertex and altitude f a plyhedrn? What is the difference between right and blique? In what case is a cne a special case f a pyramid? Is it pssible fr a sphere t have tw great circles that d nt intersect? Hw did yu cmbine cnjectures, frmulas and prperties t slve prblems in this sectin? Hw des the cncept f vlume differ frm the cncept f area? Just lking at the units, hw can yu tell if a slutin is linear r vlume r area? What wuld be a cnvincing example f the Oblique Prism- Cylinder Vlume Cnjecture? Hw des the vlume f a pyramid r cne relate t the vlume f its circumscribed prism r cylinder? Explain hw yu derived the vlume frmulas in this sectin. Lessn 10-1 Lessn 10-2 Lessn 10-3 Explre p. 544 Cnceptual Identify the Shapes f 2D crss-sectins f 3D bjects and identify 3D bjects generated by rtatins f 2D bjects (vide) Illustrative Mathematics: Use Cavalieri s Principle t Cmpare Aquarium Vlumes Prcedural Skills and Fluency: MathOpenRef: Slid Gemetry 2016-2017 10 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume Suggested Sequence f Key Events and Instructin Expectatins The area f a triangle is ½ the area f the rectangle with the same base and height. Why, then, is the area f a cne r pyramid 1/3 that f its circumscribed cylinder r prism? If the dimensins f the base f a slid are dubled and the height f that slid is dubled, by hw much is the vlume increased? Discvering Gemetry Lessns) Gemetry ACC 2-3 2-3 I will slve vlume and displacement prblems by I will investigate spheres by Cmbining understanding f area and the Pythagrean Therem with knwledge f vlume frmulas. Using displacement t calculate the vlume and density f an irregularly shaped bject. Answering questins such as Hw culd yu find the vlume f an irregularly shaped bject? What are steps that yu can take t slve a wrd prblem? What is the difference between vlume, mass, and density? What weighs mre, a pund f lead r a pund f feathers? Will heavier bjects displace mre water than lighter bjects f the same shape? Deriving the frmula fr the vlume f a sphere. Using the vlume frmulas fr a sphere and a pyramid t derive the surface area frmula fr a sphere. Applying vlume and surface area frmulas t prblems invlving spheres r hemispheres. Answering questins such as Cmpare the vlume f a sphere t the vlume f a cylinder, cne r cube with the same radius and/r height. Lessn 10-4 Lessn 10-5 Lessn 10-6 Lessn 10-7 Cnceptual Vlume Displacement and Density Activity Applicatin: Illuminatins: Cubed Cans 3-Act Lessn: Water Tank MathVisin Prject: Sand Castles 3-Act Lessn: Fill Er Up Cnceptual Vlume Frmulas (interactive) Prcedural Skills and Fluency: 2016-2017 11 Psted 2/28/17
Unit 5 Area, the Pythagrean Therem, and Vlume 1-2 2 1-2 I will check my understanding f vlume by participating in the FAL. I will prepare fr the unit assessment n area, the Pythagrean Therem, and vlume by Suggested Sequence f Key Events and Instructin Discvering Gemetry Expectatins Lessns) Des Cavalieri s Principle apply t a hemisphere? If a cne r a sphere is inside a circumscribed cylinder, what is the rati f the vlume f the cne t the vlume f the cylinder? The rati f the vlume f the sphere t the vlume f the cylinder? Think f the pattern f the leather patches n the utside f a sccer ball. Can this be used t apprximate the surface area f a sphere? If the rati f the vlume f a sphere t the vlume f a cylinder f the same radius and height is 2:3, what is the rati f their surface areas? Frmative Assessment Lessn Vlumes f Cmpund Objects Incrprating the Standards fr Mathematical Practice (SMPs) alng with the cntent standards t review the unit. Unit Assessment (LBUSD Math Intranet, Assessment) Gemetry ACC MathOpenRef: Slid Gemetry Applicatin: Illuminatins: Ice Cream Puddle 3-Act Lessn: Meatballs Cnceptual FAL: Vlumes f Cmpund Objects Prcedural Skills and Fluency: Gemetry Review 2016-2017 12 Psted 2/28/17