Common Core Georgia Performance Standards

2/23/2012Common Coreg PerformanceGeorgiaStandardsAnalytic GeometryBrooke Kline and James PrattSecondary Mathematics Specialists1

2/23/2012Thank you for being here today. You will need the following materialsduring today’s broadcast:Analytic Geometry handoutsCompass & StraightedgeScissorsNote-taking materialsActivate your brainUnderstandsimilarity in termsof similaritytransformations.Understand andapply theoremsabout circles.2

2/23/2012Why Common Core Standards? Preparation: The standards are college- and careery Theyy will helppppreparepstudents with theready.knowledge and skills they need to succeed ineducation and training after high school. Competition: The standards are internationallybenchmarked. Common standards will help ensurecompetitiveour students are globally competitive. Equity: Expectations are consistent for all – and notdependent on a student’s zip code.Why Common Core Standards? Clarity: The standards are focused, coherent, andclear Clearer standards help students (andclear.parents and teachers) understand what isexpected of them. Collaboration: The standards create a foundationto work collaboratively across states and districts,li resources andd expertise,titto createtpoolingcurricular tools, professional development,common assessments and other materials.3

2/23/2012Common Core State StandardsBuilding on the strength of current statestandards, the CCSS are designed to be: Focused, coherent, clear and rigorousInternationally benchmarkedA h d iin collegelld career readinessdiAnchoredandEvidence and research basedCommon Core State Standards in MathematicsK1234567Measurement and DataCC8Statistics and ProbabilityNumber and OperationsFractionsRatios &ProportionalRelationshipsFunctionsThe Number SystemNumber andQuantityExpressions andEquationsAl bAlgebraNumber and Operations in Base TenOperations and Algebraic Thinking9 - 12GeometryModeling Copyright 2011 Institute for Mathematics and Education4

2/23/20121. Makee sense of problemms andperseevere in solving thhem.6. Attennd to precision.Standards for Mathematical Practice2. Reason abstractly andquantitatively.3. Construct viable arguments andcritiq e the reasoning of otherscritiqueReasoning andexplaining4. Model with mathematics.5. Use appropriate toolsstrategically.Modeling andusing tools7. Look for and make use ofstructure.8. Look for and express regularity inrepeated reasoning.Seeingstructure andgeneralizing(McCallum, 2011)AlgebraCreating Equations A.CEDCreate equations that describe numbers or relationships.MCC9-12.A.CED.1 Create equations and inequalities in one variable and usethem to solve problems. Include equations arising from linear andquadratic functions, and simple rational and exponential functions. MCC9-12.A.CED.2 Create equations in two or more variables to representrelationships between quantities; graph equations on coordinate axeswith labels and scales. 5

2/23/2012While the standards focus on what is mostessential, they do not describe all that can org Aggreat deal is left to theshould be taught.discretion of teachers and curriculumdevelopers. The aim of the standards is toarticulate the fundamentals, not to set out anexhaustive list or a set of restrictions that limitsh t can beb ttaughtht bd whath t iis specified.ifi dwhatbeyondcorestandards.orgWhat’s an Analytic GeometryTeacher to do? Read your grade level standards Use the CCGPS Teaching Guide found onGeorgia Standards.org and LearningVillage Discuss the standards with yourcolleagues6

2/23/2012Analytic Geometry is the second course inthe sequence. The course embodies adiscrete study of geometry analyzed bymeans of algebraic operations withcorrelated probability/statisticspyapplications and a bridge to the thirdcourse through algebraic topics.GeometryGeometric ConstructionsSimilar TrianglesCongruent TrianglesProofsRight TriangleTrigonometryCirclesAngles and LineSegments of CirclesArea and VolumeFormulasAlgebraComplex NumbersQuadratic FunctionsSolving QuadraticEquationsProbability/StatisticsFit Quadratic Functions toDataIndependent ProbabilityConditional ProbabilityProbability of CompoundEvents7

2/23/2012Analytic Geometry OverviewUnit 1: Similarity, Congruence, and ProofsGeometryy – Similarity,y Rightg Triangles,gand Trigonometrygy Understand similarity in terms of similarity transformations Prove theorems involving similarityGeometry – Congruence Understand congruence in terms of rigid motions Prove geometric theorems Make geometric constructions8

2/23/2012Analytic Geometry OverviewUnit 2: Right Triangle TrigonometryGeometry – Similarity, Right Triangles, and Trigonometry Define trigonometric ratios and solve problems involving right trianglesAnalytic Geometry OverviewUnit 3: Circles and VolumeGeometry – Circles UnderstandU d t d andd applyl ththeorems aboutb t circlesi l Find arc lengths and areas of sectors of circlesGeometry – Geometric Measurement and Dimension Explain volume formulas and use them to solve problems9

2/23/2012Analytic Geometry OverviewUnit 4: Extending the Number SystemNumber and Quantity – The Real Number System Extend the properties of exponents to rational numbers Use properties of rational and irrational numbersNumber and Quantity – The Complex Number System Perform arithmetic operations with complex numbersAlgebra – Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomialsAnalytic Geometry OverviewUnit 5: Quadratic FunctionsNumber and Quantityy – The Complex Number Systemy Use complex numbers in polynomial identities and equationsAlgebra – Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent form to solve problemsAlgebra – Creating Equations Creating equations that describe numbers or relationshipsAlgebra – Reasoning with Equations and Inequalities Solve equations and inequalities in one variable Solve systems of equations10

2/23/2012Analytic Geometry OverviewUnit 5: Quadratic FunctionsFunctions – Interpreting Functions Interpret functions that arise in applications in terms of the context Analyze functions using different representationsFunctions – Building Functions Build a function that models a relationship between two quantities Build new functions from existing functionsFunctions – Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problemsStatistics and Probability – Interpreting Categorical and Quantitative Data Represent data on two quantitative variables on a scatter plot, and describe how thevariables are relatedAnalytic Geometry OverviewUnit 6: Modeling GeometryGeometry – Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section Use coordinates to prove simple geometric theorems algebraicallyAlgebra – Reasoning with Equations and Inequalities Solve systems of equations11

2/23/2012Analytic Geometry OverviewUnit 7: Applications of ProbabilityStatistics and Probability – Conditional Probability and the Rules of Probability Understand independence and conditional probability and them to interpret data Use the rules of probability to compute probabilities of compound events in auniform probability modelFocusCoherenceFluencyDeep UnderstandingApplicationsBalanced Approach12

2/23/2012FocusCoherenceFluencyDeep UnderstandingApplicationsBalanced ApproachFocusThe student spends more time thinking and working on priorityconcepts. is able to understand concepts and their connections toprocesses (algorithms).13

2/23/2012FocusThe teacher builds knowledge, fluency and understanding of whyand how certain mathematics concepts are done. thinks about how the concepts connect to one another. paysy more attention to priorityy content and invests theappropriate time for all students to learn before movingonto the next topic.FocusThe mile-wide inch-deep problem looks different in high school. Inearlier grades its a matter of having too many topics. In high school itsa matter of having too many separately memorized techniques, with nooverall understanding of the structure to tie them altogether. Sonarrowing and deepening the curriculum is not so much a matter ofeliminating topics, as seeing the structure that ties them together. Forexample, if students see that the distance formula and the trig identitysin 2(t) cos 2(t) 1 are both manifestations of the Pythagoreantheorem they have an understanding that helps them reconstructtheorem,these formulas rather than memorize them Bill McCallum – CCSS author14

2/23/2012Priorities in Support of Rich Instruction and ExpectationsGrade of Fluency and Conceptual UnderstandingK–2Addition and subtraction, measurement using whole number quantities3-5Multiplication and division of whole numbers and fractions6Ratios and proportional reasoning; early expressions and equations7Ratios and proportional reasoning; arithmetic of rational numbers8Linear algebra9-12ModelingModeling in Analytic GeometryWhat distinguishes modeling from other forms of applications ofmathematics are (1) explicit attention at the beginning to the process ofgetting from the problem outside of mathematics to its mathematicalformulation and (2) an explicit reconciliation between the mathematicsand the real-world situation at the end. Throughout the modelingprocess, consideration is given to both the external world and themathematics, and the results have to be both mathematically correctand reasonable in the real-world context.“The Definition of Modeling” Henry O. Pollak15

2/23/2012FAL Structure Pre-Assessment / openingpgCollaborative activityWhole-class discussionReturn to the ppre-assessment / openingpgUnderstandindependence andconditional probabilityand use them tointerpret data.Focus taskBuild a function thatmodels a relationshipbetween two quantities.16

2/23/2012FocusCoherenceFluencyDeep UnderstandingApplicationsBalanced ApproachCoherenceThe student. builds on knowledge from year to year, in a coherentlearning progression.17

2/23/2012CoherenceThe teacher. connects mathematical ideas across grade levels. thinks deeply about what is being focused on. thinks how those ideas connect to how it was taughtthe years before and the years afterafter.What do Analytic Geometrystudents bring?What are they connecting to later?Solve systemsof equations.18

2/23/2012Solving Systems Overview6th Grade Graph in all four quadrants of the coordinate plane7th Grade Solve equations of the form px q r8th Grade Solve linear equations and graph linear functions Solve simple systems of two linear equations graphicallyand algebraicallySolving Systems OverviewCoordinate Algebra SSolvel lilinear equationsti Solve systems of linear equations exactly andapproximatelyAnalytic Geometry Graph circles and parabolas Derive the equation of a circle and a parabola19

2/23/2012Solving Systems OverviewAdvanced Algebrag Solve systems consisting of linear and quadraticequationsPre-Calculus Solve systems of linear equations using matrices Solve systems of various equationsFocusCoherenceFluencyDeep UnderstandingApplicationsBalanced Approach20

2/23/2012FluencyThe student. spends time practicing skills with intensity andfrequency.FluencyThe teacher. pushes students to know skills at a greater level offluency based on understanding. focuses on the listed fluencies by grade level.21