Unit 6 Circles, Circumference, Area and Volume

Transcription

1 Unit 6 Circles, Circumference, Area and Vlume Number f Days: 38 4/10/17 6/9/17 Unit Gals Stage 1 Unit Descriptin: In Unit 6 students prve and apply basic circle therems such as: a tangent line is perpendicular t a radius therem; the inscribed angle therem; and therems abut chrds, secants, and tangents dealing with segment lengths and angle measures. They study relatinships amng segments n chrds, secants, and tangents as an applicatin f similarity. In the Cartesian crdinate system, students explain the crrespndence between the definitin f a circle and the equatin f a circle written in terms f the distance frmula, its radius, and the crdinates f its center. Given an equatin f a circle, they draw the graph in the crdinate plane, and apply techniques fr slving quadratic equatins. Students experience with twdimensinal bjects is extended t include infrmal and frmal explanatins f circumference, area and vlume frmulas. Additinally, students use their knwledge f tw dimensinal shapes t cnsider the shapes f crss sectins and the result f rtating a tw dimensinal bject abut a line. They reasn abstractly and quantitatively t develp, justify and apply vlume frmulas. Materials: Patty paper, straight edges, cmpasses, calculatrs, dynamic sftware (Desms, Gegebra) 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 Clusters Addressed [s] G-CO.A Experiment with transfrmatins in the plane. 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. UNDERSTANDINGS Students will understand that Making Meaning The arc length, sectr area, and radian measure f a circle can be fund using similarity and prprtinal reasning. There are underlying patterns and structures in the circle s prperties and pstulates. One can translate between gemetric descriptins and the algebraic equatin f a circle. New gemetric frmulas can be derived by cmbining previusly knwn frmulas. Crdinates can be used t cnstruct gemetric prfs. Gemetric transfrmatins can be expressed as functins. Gemetric shapes can be used t mdel real-wrld bjects and slve design challenges. The vlume frmula fr a prism r a cylinder invlves the area f the Base times the height. The vlume f a cne r a pyramid is ne-third that f its related cylinder r prism. ESSENTIAL QUESTIONS Students will keep cnsidering When are frmulas needed and when can yu use similarity r prprtinal reasning t find values n a circle? Which frm f the circle equatin makes it easiest t see a key feature? Hw can yu use previus frmulas t derive new frmulas? Hw wuld yu use a figure s crdinates t prve that tw gemetric shapes are cngruent? Hw wuld yu use a gemetric figure t mdel a real-wrld bject? What can yu think abut if yu dn t remember a vlume frmula? Psted 4/13/17

2 Unit 6 Circles, Circumference, Area and Vlume [s] G-CO.D Make gemetric cnstructins. [a] G-C.A Understand and apply therems abut circles. [a] G-C.B Find arc lengths and areas f sectrs f circles. [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. [a] G-GMD.B Visualize relatinships between twdimensinal and three-dimensinal bjects. [m] G-MG.A Apply gemetric cncepts in mdeling situatins. KNOWLEDGE Students will knw Unit Gals Stage 1 Acquisitin Inscribed angles with sides that intercept the end pints f a diameter are right angles. The radius f a circle is perpendicular t the tangent where the radius intersects the circle. Precise definitins f angle, circle, and arc. The frmula fr the area f a circle and, frm that, the frmula fr the area f a sectr. The radian measures f benchmark angles frm 0 t 180. The relatinships amng the central angle f a circle, inscribed angles, radii, chrds, arc lengths and sectrs. A circumscribed angle f a circle and its assciated central angle are supplementary. Gemetric shapes, measures and prperties can be used t find the distance t the hrizn. Crdinates can be used t derive gemetric frmulas. A gemetric descriptin can be used t create an algebraic representatin. As the number f sides f a regular plygn increases, the perimeter f the figure appraches the circumference f a circumscribed circle. Crdinates can be used t slve fr the area and perimeter f a figure. Three-dimensinal bjects can be generated by rtating a twdimensinal bject arund an axis. Cavalieri s Principle. Vlume frmulas fr cylinders, pyramids, cnes and spheres. Vlume frmulas fr prisms, cylinders, pyramids, and cnes are related. SKILLS Students will be skilled at and/r be able t Identify and describe the relatinships between inscribed angles, radii and chrds. Cnstruct a tangent t a circle frm a pint utside the circle. Give an infrmal argument fr the frmula fr the area f a circle. Find the length f an arc r the area f a sectr using prprtinal reasning. Cnvert frm degree measure t radian measure. Slve fr an intercepted arc r its inscribed angle. Prve the prperties f the angles f a quadrilateral inscribed in a circle. Find the measure f the angle frmed at the intersectin f tw lines tangent t a circle. Use the crdinates f the center f a circle and that circle s radius, alng with the Pythagrean Therem, t derive the frmula fr the equatin f a circle. Cmplete the square t find the center and radius f a circle given by an equatin. Derive the frmula fr the circumference f a circle. Slve fr the area f an image when given the area f a preimage and its scale factr change. Generate three-dimensinal shapes by rtating a tw-dimensinal shape arund an axis. Use Cavalieri s Principle t verify the frmulas fr prisms and cylinders. Slve fr the vlume f a prism, cylinder, pyramid, cne r sphere Psted 4/13/17

3 Unit 6 Circles, Circumference, Area and Vlume Assessed Grade Level Standards 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 [s] G-CO.A Experiment with transfrmatins in the plane. G.CO.1 Knw precise definitins f angle, circle, perpendicular line, parallel line, and line segment, based n the undefined ntins f pint, line, distance alng a line, and distance arund a circular arc. [s] G-CO.D Make gemetric cnstructins. G-CO.13 Cnstruct an equilateral triangle, a square, and a regular hexagn inscribed in a circle. [a] G-C.A Understand and apply therems abut circles. G-C.1 Prve that all circles are similar. G-C.2 Identify and describe relatinships amng inscribed angles, radii, and chrds. Include the relatinship between central, inscribed, and circumscribed angles; inscribed angles n a diameter are right angles; the radius f a circle is perpendicular t the tangent where the radius intersects the circle. G-C.3 Cnstruct the inscribed and circumscribed circles f a triangle, and prve prperties f angles fr a quadrilateral inscribed in a circle. [a] G-C.B Find arc lengths and areas f sectrs f circles. [Radian intrduced nly as unit f measure.] 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. Cnvert between degrees and radians. CA [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. [Include distance frmula; relate t Pythagrean therem] G-GPE.4 Use crdinates t prve simple gemetric therems algebraically. Fr example, prve r disprve that a figure defined by fur given pints in the crdinate plane is a rectangle; prve r disprve that the pint (1, 3) lies n the circle centered at the rigin and cntaining the pint (0, 2) Psted 4/13/17

4 Unit 6 Circles, Circumference, Area and Vlume Assessed Grade Level Standards [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.3 Use vlume frmulas fr cylinders, pyramids, cnes, and spheres t slve prblems. [a] G-GMD.B Visualize relatinships between tw-dimensinal and three-dimensinal bjects. G-GMD.4 Identify the shapes f tw-dimensinal crss-sectins f three-dimensinal bjects, and identify three-dimensinal bjects generated by rtatins f tw-dimensinal bjects. G-GMD.5 Knw that the effect f a scale factr k greater than zer n length, area and vlume is t multiply each by k, k 2, k 3, respectively; determine length, area and vlume measures using scale factrs. CA [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). G-MG.3 Apply gemetric methds t slve design prblems (e.g., designing an bject r structure t satisfy physical cnstraints r minimize cst; wrking with typgraphic grid systems based n ratis). 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 Psted 4/13/17

5 Unit 6 Circles, Circumference, Area and Vlume Assessment Evidence Unit Assessment Evidence f Learning Stage 2 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 Claim1: [s] G-CO.A Students will knw the precise definitin f a circle. Students will knw the precise definitin f the distance arund a circular arc. [s] G-CO.D Students will cnstruct an equilateral triangle, a square, and a regular hexagn inscribed in a circle. [a] G-C.A Students will prve that all circles are similar. Students will identify and describe relatinships amng inscribed angles, radii and chrds. Students will cnstruct the inscribed and circumscribed circles f a triangle. Students will prve prperties f angles fr a quadrilateral inscribed in a circle. [a] G-C.B Students will derive using similarity the fact that the length f the arc intercepted by an angle is prprtinal t the radius. Students will define the radian measure f the angle as the cnstant f prprtinality. Students will derive the frmula fr the area f a sectr. Students will cnvert between degrees and radians. [a] G-GPE.A Students will derive the equatin f a circle f a given center and radius using the Pythagrean Therem. Students will cmplete the square t find the center and radius f a circle given by an equatin. [m] G-GPE.B Students will use crdinates t prve simple gemetric therems algebraically, such as whether r nt a given pint lies n the circle centered at the rigin and cntaining a pint P. [a] G-GMD.A Students will give infrmal arguments fr the frmulas fr the circumference f a circle, area f a circle, vlume f a cylinder, pyramid, and cne. Students will give infrmal arguments fr frmulas using dissectin arguments, Cavalieri s principle and infrmal limit arguments. Students will use vlume frmulas fr cylinders, pyramids, cnes, and spheres t slve prblems Psted 4/13/17

6 Unit 6 Circles, Circumference, Area and Vlume Evidence f Learning Stage 2 Assessment Evidence [a] G-GMD.B Students will identify the shapes f tw-dimensinal crss-sectins f three-dimensinal bjects. Students will identify three-dimensinal bjects generated by rtatins f tw-dimensinal bjects. Students will knw that the effect f a scale factr k greater than zer n length, area, and vlume is t multiply each by k, k 2, k 3 respectively. Students will determine length, area, and vlume measures using scale factrs. [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 t mdeling situatins. Students will apply gemetric methds t slve design prblems. 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: Nne Other Evidence 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: G-CO.A 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 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 Perfrmance Tasks SBAC Interim Assessment Blcks Access Using Frmative Assessment fr Differentiatin fr suggestins. Lcated n the LBUSD website M Mathematics Curriculum Dcuments Psted 4/13/17

7 Unit 6 Circles, Circumference, Area and Vlume Suggested Sequence f Key Learning Events and Instructin Days Learning Target Expectatins Lessns) I will explre the Opening Task- Where t Put the Brand New Speakers prperties f a circle by participating in the Given the utline f a family rm and the placement f three Opening Task: speakers, students will decide n where a chair shuld be placed fr Where t Put the best appreciatin f the sund frm their TV. They will sketch a chair 1-2 Brand New at their chsen lcatin and explain that placement. Their chice Speakers. and explanatin will be supprted with mathematical language. Applicatins: Where t Put the Brand New Speakers 4-5 I will investigate special segments and slve fr angle and arc measures by Revisit this task after the vlume prtin f Unit 6. Encurage the students t think f ther variables that might affect the placement f their speakers. Identifying special segments and lines. Drawing and identifying cmmn tangents. Using prperties f tangents. Finding arc measures Identifying cngruent arcs. Prving that all circles are similar. Using chrds f circles t find lengths and arc measures. Answering questins such as What d yu already knw abut circles? Are all chrds diameters? Are all diameters chrds? Why is it necessary t state that a tangent is in the same plane as a circle? In hw many ways can tw circles intersect? What are the prperties f a line which is tangent t a circle? Hw des ntatin differ based n the gemetric shape being referred t (ex.: circle r majr and minr arc)? Hw d yu knw if tw arcs are cngruent? Circles? Hw d yu knw if tw circles are similar? Are tw arcs with the same measure similar r cngruent? Hw d yu knw if tw chrds are cngruent? If tw chrds f a circle are perpendicular and cngruent, des ne f them have t be a diameter? Sectin 10.1 Sectin 10.2 Sectin 10.3 Cnceptual Illuminatins: Flding Circles: Explring Circle Therems thrugh Paper Flding Prcedural Skills and Fluency: Math Open Ref: Circles and Ellipses Applicatins: 3-Act Lessn: Rtunda West Flrida Psted 4/13/17

8 Unit 6 Circles, Circumference, Area and Vlume Suggested Sequence f Key Learning Events and Instructin Days Learning Target Expectatins Lessns) I will investigate Using inscribed angles. Sectin 10.4 angle relatinships in Using inscribed plygns. Sectin 10.5 circles and plygns Finding angle and arc measures. by Using circumscribed angles. Answering questins such as Can every triangle be circumscribed by a circle? Can every quadrilateral be circumscribed by a circle? The diagnals f a rectangle inscribed in a circle are the diameters f that circle. Why? Hw d yu summarize the three cases fr finding the angle measure f the angle frmed by tw intersecting lines and a circle? Hw d yu find the measure f a circumscribed angle? 4-5 Cnceptual Illustrative Mathematics: Right Triangles Inscribed in Circles 1 Illustrative Mathematics: Right Triangles Inscribed in Circles 2 Illustrative Mathematics: Inscribing a Square in a Circle MathematicsVisin Prject: Mre Things Under Cnstructin Prcedural Skills and Fluency: Math Open Ref: Inscribed Angles Math Open Ref: Interir Angles f an Inscribed Quadrilateral Khan Academy: Cnstructing a Line Tangent t a Circle Psted 4/13/17

9 Unit 6 Circles, Circumference, Area and Vlume Suggested Sequence f Key Learning Events and Instructin Days Learning Target Expectatins Lessns) I will investigate segment relatinships and equatins f circles by 4-5 Using segments f chrds, tangents, and secants. Writing and graphing equatins f circles. Writing crdinate prfs invlving circles. Slving real-life prblems using graphs f circles. Answering questins such as What steps wuld yu take t prve segment relatinships in circles? Will using any pint n the circle result in the same standard equatin fr the circle? What are the steps fr cmpleting the square? Why is knwing the lcatin f the center and ne pint n a circle enugh t graph a circle? Sectin 10.6 Sectin 10.7 STEM Vide: Seismgraphs and Earthquake Epicenters Cnceptual MathOpenRef: Crdinate Desms: Circle Patterns Desms: Lawnmwer Math Applicatins: STEM Vide Perfrmance Task: Finding Lcatins 2-3 I will check my ability t use use crdinates t slve fr the equatin f a circle by participating in the FAL. FORMATIVE ASSESSMENT LESSON Equatins f Circles 1 Cnceptual FAL: Equatins f Circles I Psted 4/13/17

10 Unit 6 Circles, Circumference, Area and Vlume 4-5 Suggested Sequence f Key Learning Events and Instructin Days Learning Target Expectatins Lessns) I will investigate the characteristics f circles and plygns by Using the frmula fr circumference. Using arc lengths t find measures. Slving real-life prblems. Measuring angles in radians. Finding the areas f rhmbuses and kites. Finding angle measures in regular plygns. Finding areas f regular plygns. Answering questins such as: What type f number is π? What des irratinal mean? Hw is arc measure different frm arc length? When yu find an answer fr a prblem cntaining π, shuld yu leave yur answer in terms f π r use a decimal apprximatin? Withut using a frmula, hw can yu find the area f a sectr f a circle? If an arc measure is dubled, will the area f the sectr als be dubled? What are the attributes f the diagnals f a rhmbus? A kite? What infrmatin is needed t find the areas f regular plygns? What happens t the area f a kite if yu duble the length f a diagnal? Bth diagnals? What are varius methds t find the area f a regular plygn? Sectin 11.1 Sectin 11.2 STEM Vide: Ppulatin Density Sectin 11.3 Cnceptual MathOpenRef: Radians MathOpenRef: Area f a Regular Plygn MathOpenRef: Area f a Rhmbus MathOpenRef: Area f a Kite Prcedural Skills and Fluency: Quizlet: Degrees t Radians Flashcards Applicatins: STEM Vide Perfrmance Task: Ppulatin Density Psted 4/13/17

11 Unit 6 Circles, Circumference, Area and Vlume 5-6 Suggested Sequence f Key Learning Events and Instructin Days Learning Target Expectatins Lessns) I will investigate the Classifying slids. Sectin 11.4 vlumes f slids Describing crss sectins. Sectin 11.5 by Sketching and describing slids f revlutin. Sectin 11.6 Finding vlumes f prisms and cylinders. Using the frmula fr density. Using vlumes f prisms and cylinders. Finding vlumes f pyramids. Using vlumes f pyramids. Answering questins such as: Hw many crss-sectinal shapes can be made when yu slice thrugh a cube? What tw-dimensinal figures result when a crss-sectin is taken frm a three-dimensinal figure? Des the Cavalieri Principle apply t just rectangular prisms? What is the relatinship between the numbers f vertices, edges and faces f a plyhedrn? Why did Plat say that there are precisely five Platnic Slids? What tw-dimensinal figures, rtated arund an axis, generate three-dimensinal shapes Hw can yu find the vlume f a prism r cylinder that is nt a right prism r right cylinder? Just by lking at the units, hw d yu knw if the prblem is asking abut area r vlume? Is the frmula fr the vlume f a pyramid similar t the ther vlume frmulas? Hw des the vlume f a pyramid with a square base change when the height stays the same and the side length f the base is dubled? Is that true fr all square pyramids? Cnceptual MathOpenRef: Slid Vlume Displacement and Density Activity 3-Act Lessn: Yu Pur, I Chse Prcedural Skills and Fluency: Identify the Shapes f 2D crss-sectins f 3D bjects and identify 3D bjects generated by rtatins f 2D bjects (vide) Vlume Frmulas (interactive) Applicatins: The Open-Tpped Bx The Open-Tpped Bx Supprt Blg TI-84 Cmmands fr Scatter Plts Illuminatins: Fishing fr the Best Prism 3-Act Lessn: Meatballs 3-Act Lessn: Water Tank Psted 4/13/17

12 Unit 6 Circles, Circumference, Area and Vlume 3-4 Suggested Sequence f Key Learning Events and Instructin Days Learning Target Expectatins Lessns) I will investigate the Finding surface areas f right cnes. Sectin 11.7 surface area and Finding vlumes f cnes. Sectin 11.8 vlume f cnes and Using vlumes f cnes. spheres by Finding surface areas f spheres. Finding vlumes f spheres. Answering questins such as: Hw des a net help yu understand hw t find the surface area f a three-dimensinal shape? Hw are the vlume frmulas fr pyramids and cnes similar? Different? Des Cavalieri s Principle hld true fr cnes? What are the main steps in develping the frmula fr the vlume f a sphere? If yu knw the circumference f a sphere, can yu calculate its surface area? If the radius f a sphere is dubled, will the surface area als duble? Cnceptual MathOpenRef: Slid Which One Desn t Belng: Circles, Circumference, Area, and Vlume Applicatins: Illuminatins: Ice Cream Puddle Hw Many Gumballs Fit in the Gumball Machine? 3-Act Lessn: Meatballs 2-3 I will check my ability t use the vlume f bjects t slve fr the vlume f a cmpund bject by participating in the FAL. Frmative Assessment Lessn Calculating Vlumes f Cmpund Objects Cnceptual Calculating Vlumes f Cmpund Objects Psted 4/13/17

13 Unit 6 Circles, Circumference, Area and Vlume Days Learning Target Expectatins 2-3 I will prepare fr the unit assessment circles, circumference, area and vlume by... Suggested Sequence f Key Learning Events and Instructin Incrprating the Standards fr Mathematical Practice (SMPs) alng with the cntent standards t review the unit. Lessns) Prcedural Skills and Fluency: Jepardy: Circle Review MathematicsVisin Prject: Cnstructin Blueprints 1-2 Unit Assessment (LBUSD Math Intranet, Assessment) Applicatins: 3-Act Lessn: Lucky Cw Psted 4/13/17