Curse Syllabus MATH 205: Gemetry fr the Middle Level Teacher Catalg Descriptin with Prerequisites This curse is designed t equip middle level (4-8) mathematics specialists with sufficient knwledge and mathematical experiences fr teaching gemetry and measurement effectively. The curse includes the study f tw-dimensinal and three-dimensinal figures, gemetric cnstructins, cngruence, similarity, angle measure, distance, area and vlume. Cnnectins between gemetry and ther mathematics tpics, nature and art are addressed. Prerequisite: passing scre n BST, and C r better in MATH 104 r department permissin The need fr a gemetry curse fr middle level teachers is justified by the majr curricular develpments in middle schl mathematics. Current recmmendatins fr the middle schl mathematics curriculum stress the critical rle that middle schl plays as a transitin pint in develping the ideas f gemetry and measurement frm elementary t secndary grades. Tpics t be taught in gemetry and measurement in middle schl include: Develp prperties f tw- and three-dimensinal gemetric shapes, Prmte mathematical arguments prving r disprving gemetric relatinships amng gemetric figures, Apply transfrmatins and symmetry t analyze mathematical situatins, Develp spatial reasning skills and gemetric mdeling t slve prblems, Identify measurable quantities, apprpriate units and prcesses f measurement, Apply apprpriate tls, techniques, and frmulas t determine measurements (NCTM, 2000). Subsequently, the mathematical cntent preparatin fr middle level teachers must be f a qualitatively different nature than fr elementary and secndary teachers. Currently, the Mathematics Department ffers the curse Fundamentals f Mathematics II [MATH105] which includes units n gemetry and measurement. MATH 105 will be retained in its current frm, with minr revisins, s that it will better serve the mathematical cntent needs fr Early Childhd [PK 4] prspective teachers. As a result, while there will be sme verlap f tpics with MATH 105, there is n cmparable curse ffered frm ther departments at Millersville University, due t the rigr f the develpment f the cntent, and the instructinal methds used t teach the curse. The primary fcus f the curse is t prvide students with the cntent knwledge and learning experiences necessary t teach middle level gemetry and measurement as recmmended by the Pennsylvania Department f Educatin Academic Standards fr Mathematics, the Pennsylvania Mathematics Assessment Anchrs, and the NCTM Principles and Standards fr Schl Mathematics. Students will study the tpics f tw- and threedimensinal shapes, cngruence, and similarity frm the multiple perspectives f synthetic, analytic and transfrmatinal gemetry. Students will study the meaning f fundamental units f measurement and their frmulas, determine apprpriate units fr each attribute, sharpen their skill at using relevant tls and measurement prcesses, and apply their measurement skills t slve prblems. Students will be engaged in learning thrugh cmpleting lab activities using bth physical manipulatives [e.g. paper-flding, MIRA] and current dynamic gemetry sftware [e.g. Gemeter s Sketchpad, Gegebra]. Students will explre the cnnectins between gemetry and tpics the nature [e.g. symmetry], art and design [e.g. tessellatins, wallpaper patterns]. 1
Primary Curse Objectives Upn cmpletin f the curse, students will be able t: Objective 1. Apply fundamental rules f lgic t frm cherent written and ral mathematical arguments 2. Perfrm standard gemetric cnstructins using paper flding, cmpass & Frms f Assessment Exams, hmewrk sets and Exams and straightedge, MIRA, and dynamic gemetry sftware 3. Identify, define, and apply prperties f tw-and three-dimensinal figures Exams and hmewrk sets 4. Apply rigid mtins t the study f cngruent figures Exams, hmewrk sets and 5. Apply dilatins and ther size transfrmatins t the study f similar figures Exams, hmewrk sets and 6. Apply measurement techniques and frmulas t gemetric figures Exams, hmewrk sets and 7. Demnstrate spatial reasning skills in tw- and three dimensinal settings Lab exercises 8. Demnstrate/apply cnnectins f gemetric tpics t ther disciplines Lab exercises In additin t the abve curse bjectives, MATH 205 will help teacher candidates at the Middle Level t fulfill university, state, and natinal standards as indicated belw. I. In alignment with Millersville University s Prfessinal Educatin Unit s Cnceptual Framewrk, this curse will serve t satisfy key indicatrs fr Prficiency Area I: Cntent Knwledge. Candidates will display knwledge f the mathematical cntent and apply the imprtant principles and cncepts delineated in prfessinal, state, and institutinal standards. 1Specifically candidates will: Demnstrate cmpetency in their chsen cntent area. Engage in inquiry in their cntent area that develps their ability t extend student understanding beynd surface infrmatin. Understand, explain, and apply knwledge f the cntextual issues (e.g., plitical, scial, cultural, ethnicity, disability, and gender) that influence their cntent area. Recgnize varius theries and pints f view within their field. Develp curricula in a variety f instructinal frmats reflective f state, natinal, and lcal standards. II. PDE Standards that teacher candidates will be better prepared t help their wn students t achieve after cmpleting this curse include the fllwing: PDE 2.3.8 A: Develp frmulas and prcedures fr determining measurements (e.g., area, vlume, distance). PDE 2.3.8 B: Slve rate prblems (e.g., rate time = distance, principal interest rate = interest). PDE 2.3.8 C: Measure angles in degrees and determine relatins f angles. PDE 2.3.8 D: Estimate, use and describe measures f distance, rate, perimeter, area, vlume, weight, mass and angles. PDE 2.3.8 E: Describe hw a change in linear dimensin f an bject affects its perimeter, area and vlume. PDE 2.3.8 F: Use scale measurements t interpret maps r drawings. PDE 2.5.8 A: Justify strategies and defend appraches used and cnclusins reached. PDE 2.5.8 B: Determine pertinent infrmatin in prblem situatins and whether any further 2
infrmatin is needed fr slutin. PDE 2.9.8 A: Cnstruct figures incrprating perpendicular and parallel lines, the perpendicular bisectr f a line segment and an angle bisectr using cmputer sftware. PDE 2.9.8 B: Draw, label, measure and list the prperties f cmplementary, supplementary and vertical angles. PDE 2.9.8 C: Classify familiar plygns as regular r irregular up t a decagn. PDE 2.9.8 D: Identify, name, draw and list all prperties f squares, cubes, pyramids, parallelgrams, quadrilaterals, trapezids, plygns, rectangles, rhmbi, circles, spheres, triangles, prisms and cylinders. PDE 2.9.8 E: Cnstruct parallel lines, draw a transversal and measure and cmpare angles frmed (e.g., alternate interir and exterir angles). PDE 2.9.8 F: Distinguish between similar and cngruent plygns. PDE 2.9.8 G: Apprximate the value f π (pi) thrugh experimentatin. PDE 2.9.8 H: Use simple gemetric figures (e.g., triangles, squares) t create, thrugh rtatin, transfrmatinal figures in three dimensins. PDE 2.9.8 I: Generate transfrmatins using cmputer sftware. PDE 2.9.8 J: Analyze gemetric patterns (e.g., tessellatins, sequences f shapes) and develp descriptins f the patterns. PDE 2.9.8 K: Analyze bjects t determine whether they illustrate tessellatins, symmetry, cngruence, similarity and scale. III. Cntent Knwledge Standard fr mathematics teachers frm Natinal Cuncil f Teachers f Mathematics fr the Natinal Cuncil fr the Accreditatin f Teacher Educatrs (2003). Standard 11: Knwledge f Gemetries: Candidates use spatial visualizatin and gemetric mdeling t explre and analyze gemetric shapes, structures, and their prperties. Indicatrs 11.1 Demnstrate knwledge f cre cncepts and principles f Euclidean and nn-euclidean gemetries in tw and three dimensins frm bth frmal and infrmal perspectives. 11.2 Exhibit knwledge f the rle f aximatic systems and prfs in gemetry. 11.3 Analyze characteristics and relatinships f gemetric shapes and structures. 11.4 Build and manipulate representatins f tw- and three- dimensinal bjects and visualize bjects frm different perspectives. 11.5 Specify lcatins and describe spatial relatinships using crdinate gemetry, vectrs, and ther representatinal systems. 11.6 Apply transfrmatins and use symmetry, similarity, and cngruence t analyze mathematical situatins. 11.7 Use cncrete mdels, drawings, and dynamic gemetric sftware t explre gemetric ideas and their applicatins in real-wrld cntexts. 11.8 Demnstrate knwledge f the histrical develpment f Euclidean and nn-euclidean gemetries including cntributins frm diverse cultures. Standard 15: Knwledge f Measurement: Candidates apply and use measurement cncepts and tls. Indicatrs 15.1 Recgnize the cmmn representatins and uses f measurement and chse tls and units fr measuring. 15.2 Apply apprpriate techniques, tls, and frmulas t determine measurements and their applicatin in a variety f cntexts. 15.3 Cmpletes errr analysis thrugh determining the reliability f the numbers btained frm measures. 15.4 Demnstrate knwledge f the histrical develpment f measurement and measurement systems including cntributins frm diverse cultures. Cmprehensive Outline f Curse Cntent 3
Lgic and deductive reasning (2 weeks) a. Mathematical definitins b. Types f statements: axim, therem, prpsitin, lemma, crllary, etc. c. If then statements; hypthesis, cnclusin d. Cnverse, cntrapsitive, cunterexample e. Applying the rules f lgic f. Writing mathematical prfs 2-D figures (4 weeks) a. Definitin, prperties, and classificatin f angles, triangles, quadrilaterals, ther plygns, and circles b. Using crdinate gemetry t describe figures c. Cngruent triangles d. Length, angle measure, area: cncept and applicatins e. Relevant gemetric cnstructins 3-D figures (3 weeks) a. Definitin and prperties f plyhedrns, prisms, pyramids, spheres, etc. b. Surface area and vlume: cncept and applicatins c. Relevant gemetric cnstructins d. Spatial reasning Rigid mtins (3 weeks) a. Translatin, reflectin, rtatin, and glide reflectin b. Cngruence c. Symmetry d. Tessellatins e. Rigid mtins in space f. Relevant gemetric cnstructins g. Symmetry and tessellatins in art and design Similarity (3 weeks) a. Dilatins and ther types f similarities b. Similar triangles, plygns, and ther figures c. Pythagrean therem d. Rati and prprtin e. Relevant gemetric cnstructins Vectrs and Matrices (ptinal) a. Crdinate representatin f vectrs b. Transfrmatins using matrices Nn-Euclidean gemetries (ptinal) a. Taxicab gemetry b. Spherical gemetry c. Hyperblic gemetry Criteria fr Evaluating Student Perfrmance Evaluatin f student learning will be accmplished by the fllwing: Tw r three in-class exams A cmprehensive final exam Lab exercises in and utside f class invlving: Cmpass and straightedge 4
Paper-flding MIRA r ther reflective tl Cmputer sftware such as Gemeter s Sketchpad r Gegebra Hmewrk sets cmpleted utside f class Textbks: Beem, Jhn K. (2006). Gemetry Cnnectins: Mathematics fr Middle Schl Teachers. Upper Saddle River, NJ: Pearsn Prentice Hall. Aichele, Duglas B. and Wlfe, Jhn (2008). Gemetric Structures: An Inquiry-Based Apprach fr Prspective Elementary and Middle Schl Teachers. Upper Saddle River, NJ: Pearsn Prentice Hall. General Educatin Credit This curse will satisfy G2 credit fr MATH. Resurces Cnnected Mathematics Prject (CMP). Pearsn Educatin, Inc/Michigan State University. http://cnnectedmath.msu.edu/. Gemetric Structures fr Elementary Teachers (GeSET). Oklahma State University. http://www.math.kstate.edu/geset/. Mathematics in Cntext (MiC). Encyclpedia Brittanica/University f Wiscnsin. http://shwmecenter.missuri.edu/shwme/mic.shtml. MathScape. Glence/McGraw-Hill/Educatinal Develpment Center. http://www2.edc.rg/mathscape/. MathThematics. McDugal Littlel/University f Mntana. The Gemeter s Sketchpad Resurce Center. KCP Technlgies. http://www.dynamicgemetry.cm/. 5