Common Core Georgia Performance Standards · Building on the strength of current state ... •...

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Common Core Georgia Performance g

StandardsAnalytic Geometry

Brooke Kline and James PrattBrooke Kline and James PrattSecondary Mathematics Specialists

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Thank you for being here today.

You will need the following materials during today’s broadcast:

• Analytic Geometry handouts• Compass & Straightedge • Scissors• Scissors• Note-taking materials

Activate your brain

Understand similarity in termssimilarity in terms of similarity transformations.

Understand and apply theoremsapply theorems about circles.

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Why Common Core Standards?• Preparation: The standards are college- and career-

ready. They will help prepare students with the y y p p pknowledge and skills they need to succeed in education and training after high school.

• Competition: The standards are internationally benchmarked. Common standards will help ensure our students are globally competitiveour students are globally competitive.

• Equity: Expectations are consistent for all – and not dependent on a student’s zip code.

Why Common Core Standards?• Clarity: The standards are focused, coherent, and

clear Clearer standards help students (andclear. Clearer standards help students (and parents and teachers) understand what is expected of them.

• Collaboration: The standards create a foundation to work collaboratively across states and districts,

li d ti t tpooling resources and expertise, to create curricular tools, professional development, common assessments and other materials.

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Common Core State Standards

Building on the strength of current state standards, the CCSS are designed to be:

• Focused, coherent, clear and rigorous• Internationally benchmarked

A h d i ll d di• Anchored in college and career readiness • Evidence and research based

Common Core State Standards in Mathematics

K 1 2 3 4 5 6 7 8 9 - 12

Measurement and Data Statistics and Probability

The Number SystemNumber and Operations in Base Ten

Number and Operations Fractions

Expressions andAl b

Number and Quantity

FunctionsRatios &

Proportional Relationships

CC

Modeling

Geometry

Operations and Algebraic ThinkingExpressions and

EquationsAlgebra

© Copyright 2011 Institute for Mathematics and Education

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Standards for Mathematical Practicem

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2. Reason abstractly and quantitatively.3. Construct viable arguments and critiq e the reasoning of others

Reasoning and explaining

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critique the reasoning of others

4. Model with mathematics.5. Use appropriate tools strategically.

7. Look for and make use of

Modeling and using tools

Seeing

(McCallum, 2011)

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structure.8. Look for and express regularity in repeated reasoning.

Seeing structure and generalizing

Algebra

Creating Equations★ A.CED

Create equations that describe numbers or relationships.

MCC9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

MCC9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

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While the standards focus on what is most essential, they do not describe all that can or should be taught. A great deal is left to the g gdiscretion of teachers and curriculum developers. The aim of the standards is to articulate the fundamentals, not to set out an exhaustive list or a set of restrictions that limits

h t b t ht b d h t i ifi dwhat can be taught beyond what is specified. corestandards.org

What’s an Analytic Geometry Teacher to do?

• Read your grade level standards• Use the CCGPS Teaching Guide found on

Georgia Standards.org and Learning Village

• Discuss the standards with your• Discuss the standards with your colleagues

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Analytic Geometry is the second course in the sequence. The course embodies a discrete study of geometry analyzed by means of algebraic operations with correlated probability/statistics p yapplications and a bridge to the third course through algebraic topics.

Geometry Algebra Probability/Statistics

Geometric Constructions Complex Numbers Fit Quadratic Functions to Similar TrianglesCongruent TrianglesProofsRight Triangle TrigonometryCirclesAngles and Line

Quadratic FunctionsSolving Quadratic Equations

DataIndependent ProbabilityConditional ProbabilityProbability of Compound Events

Segments of CirclesArea and Volume Formulas

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Analytic Geometry OverviewUnit 1: Similarity, Congruence, and Proofs

Geometry – Similarity, Right Triangles, and Trigonometryy y g g g y

Understand similarity in terms of similarity transformations

Prove theorems involving similarity

Geometry – Congruence

Understand congruence in terms of rigid motions

Prove geometric theorems

Make geometric constructions Make geometric constructions

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Analytic Geometry OverviewUnit 2: Right Triangle Trigonometry

Geometry – Similarity, Right Triangles, and TrigonometryGeometry Similarity, Right Triangles, and Trigonometry

Define trigonometric ratios and solve problems involving right triangles

Analytic Geometry OverviewUnit 3: Circles and Volume

Geometry – Circles U d t d d l th b t i l Understand and apply theorems about circles Find arc lengths and areas of sectors of circlesGeometry – Geometric Measurement and Dimension Explain volume formulas and use them to solve problems

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Analytic Geometry OverviewUnit 4: Extending the Number System

Number and Quantity – The Real 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 polynomials

Analytic Geometry OverviewUnit 5: Quadratic Functions

Number and Quantity – The Complex Number Systemy y

Use complex numbers in polynomial identities and equations

Algebra – Seeing Structure in Expressions

Interpret the structure of expressions

Write expressions in equivalent form to solve problems

Algebra – Creating Equations

Creating equations that describe numbers or relationshipsCreating equations that describe numbers or relationships

Algebra – Reasoning with Equations and Inequalities

Solve equations and inequalities in one variable

Solve systems of equations

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Analytic Geometry OverviewUnit 5: Quadratic Functions

Functions – Interpreting Functions

Interpret functions that arise in applications in terms of the context

Analyze functions using different representations

Functions – Building Functions

Build a function that models a relationship between two quantities

Build new functions from existing functions

Functions – Linear, Quadratic, and Exponential Models

Construct and compare linear, quadratic, and exponential models and solve problems

Statistics and Probability – Interpreting Categorical and Quantitative Data

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related

Analytic Geometry OverviewUnit 6: Modeling Geometry

Geometry Expressing Geometric Properties with EquationsGeometry – Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section

Use coordinates to prove simple geometric theorems algebraically

Algebra – Reasoning with Equations and Inequalities

Solve systems of equations

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Analytic Geometry OverviewUnit 7: Applications of Probability

Statistics and Probability Conditional Probability and the Rules 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 a uniform probability model

FocusFocusCoherenceFluencyDeep UnderstandingApplicationsApplicationsBalanced Approach

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Focus

The student… • spends more time thinking and working on priority

concepts.

• is able to understand concepts and their connections to processes (algorithms).

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Focus

The teacher…• builds knowledge, fluency and understanding of why

and how certain mathematics concepts are done.

• thinks about how the concepts connect to one another.

• pays more attention to priority content and invests the y yappropriate time for all students to learn before moving onto the next topic.

FocusThe mile-wide inch-deep problem looks different in high school. In earlier grades its a matter of having too many topics. In high school its a matter of having too many separately memorized techniques, with no overall understanding of the structure to tie them altogether. So narrowing and deepening the curriculum is not so much a matter of eliminating topics, as seeing the structure that ties them together. For example, if students see that the distance formula and the trig identity sin^2(t) +cos^2(t) = 1 are both manifestations of the Pythagorean theorem they have an understanding that helps them reconstructtheorem, they have an understanding that helps them reconstruct these formulas rather than memorize them…

Bill McCallum – CCSS author

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GradePriorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding

K–2 Addition and subtraction, measurement using whole number quantities

3-5 Multiplication and division of whole numbers and fractions

6 Ratios and proportional reasoning; early expressions and equations

7 Ratios and proportional reasoning; arithmetic of rational numbers

8 Linear algebra

9-12 Modeling

Modeling in Analytic GeometryWhat distinguishes modeling from other forms of applications of mathematics are (1) explicit attention at the beginning to the process ofmathematics are (1) explicit attention at the beginning to the process of getting from the problem outside of mathematics to its mathematical formulation and (2) an explicit reconciliation between the mathematics and the real-world situation at the end. Throughout the modeling process, consideration is given to both the external world and the mathematics, and the results have to be both mathematically correct and reasonable in the real-world context.

“The Definition of Modeling” Henry O. Pollak

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FAL Structure

• Pre-Assessment / openingp g

• Collaborative activity

• Whole-class discussion

• Return to the pre-assessment / openingp p g

Focus taskUnderstand independence and conditional probability and use them to interpret data.

Build a function that models a relationship between two quantities.

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Coherence

The student...• builds on knowledge from year to year, in a coherent

learning progression.

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Coherence

The teacher...• connects mathematical ideas across grade levels.

• thinks deeply about what is being focused on.

• thinks how those ideas connect to how it was taught the years before and the years afterthe years before and the years after.

What do Analytic Geometry students bring?

What are they connecting to later?Solve systems of equations.

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Solving Systems Overview

6th Grade• Graph in all four quadrants of the coordinate plane

7th Grade• Solve equations of the form px + q = r

8th Grade• Solve linear equations and graph linear functions• Solve simple systems of two linear equations graphically

and algebraically

Solving Systems OverviewCoordinate Algebra

S l li ti• Solve linear equations• Solve systems of linear equations exactly and

approximately

Analytic Geometry• Graph circles and parabolas• Graph circles and parabolas• Derive the equation of a circle and a parabola

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Solving Systems Overview

Advanced Algebrag• Solve systems consisting of linear and quadratic

equations

Pre-Calculus• Solve systems of linear equations using matrices• Solve systems of various equations


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Fluency

The student...• spends time practicing skills with intensity and

frequency.

Fluency

The teacher...• pushes students to know skills at a greater level of

fluency based on understanding.

• focuses on the listed fluencies by grade level.

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Grade Required Fluency

K Add/subtract within 51 Add/subtract within 102 Add/subtract within 20 & Add/subtract within 100 (pencil and paper)3 Multiply/divide within 100 & Add/subtract within 10003 p y &4 Add/subtract within 1,000,0005 Multi-digit multiplication6 Multi-digit division & Multi-digit decimal operations7 Solve px + q = r, p(x + q) = r8 Solve simple 22 systems by inspection

Algebraic manipulation in which to understand structure.W iti l t t l ti hi b t t titi

9-12Writing a rule to represent a relationship between two quantities.Seeing mathematics as a tool to model real-world situations.Understanding quantities and their relationships.

What does Fluency Look Like in Analytic Geometry?

• FLEXIBLY

• ACCURATELY

• EFFICIENTLY

• APPROPRIATELY

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Deep Understanding

The student...• shows mastery of material at a deep level in numerous

ways.

• uses mathematical practices to demonstrate understanding of different material and concepts.

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Deep Understanding

The teacher...• asks what mastery/proficiency really looks like and

means.• plans for progression of levels of understanding. • spends the time to gain the depth of the

understanding.g• becomes flexible and comfortable in own depth of

content knowledge.

What does depth mean in Analytic Geometry?

Write expressions in equivalent f t l blforms to solve problems.

Solve equations and inequalities in one variable.

Analyze functions using different representations.

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FocusFocusCoherenceFluencyDeep UnderstandingApplicationsppBalanced Approach

Application

The student...• applies mathematics in other content areas and

situations.

• chooses the right mathematics concept to solve a problem when not necessarily prompted to do so.

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Application

The teacher...• contextualizes mathematics.

• creates real world experiences in which students use what they know, and in which they are not necessarily prompted to use mathematics.

Mathematizing Analytic GeometryTranslate between the geometric gdescription and the equation for a conic section.

Extend the properties of exponents to rational exponents.

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FocusCoherenceFluencyDeep UnderstandingApplicationsApplicationsBalanced Approach

Balanced Approach

The student...• practices mathematics skills to achieve fluency.

• practices mathematics concepts to ensure application in novel situations.

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Balanced Approach

The teacher...• finds the balance between understanding and practice.

• normalizes the productive struggle.

• ritualizes skills practice.

Balanced Approach

Build a functionBuild a functionthat models a relationship between two quantities.

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What’s in Analytic Geometry B/ Advanced Algebra

• Extending the Number System◦ Extend the properties of exponents to rational exponents

Use properties of rational and irrational numbers◦ Use properties of rational and irrational numbers◦ Perform arithmetic operations with complex numbers◦ Perform arithmetic operations on polynomials

• Quadratic Functions◦ Use complex numbers in polynomial identities and equations◦ Interpret the structure of expressions◦ Write expressions in equivalent forms to solve problems◦ Analyze functions using different representationsAnalyze functions using different representations◦ Build a function that models a relationship between two quantities◦ Build new functions from existing functions◦ Construct and compare linear, quadratic, and exponential models and solve problems◦ Summarize, represent, and interpret data on two categorical and quantitative variables

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• Modeling Geometry◦ Translate between the geometric description and the equation for a conic section

Use coordinates to prove simple geometric theorems algebraically◦ Use coordinates to prove simple geometric theorems algebraically• Applications of Probability

◦ Understand independence and conditional probability and use them to interpret data◦ Use the rules of probability to compute probabilities of compound events in a uniform

probability model• Inferences and Conclusions from Data

◦ Summarize, represent, and interpret data on a single count or measurement variable◦ Understand and evaluate random processes underlying statistical experimentsUnderstand and evaluate random processes underlying statistical experiments◦ Make inferences and justify conclusions from sample surveys, experiments, and

observational studies


• Polynomial Functions◦ Use complex numbers in polynomial identities and equations◦ Interpret the structure of expressions◦ Write expressions in equivalent forms to solve problems◦ Perform arithmetic operations on polynomials◦ Understand the relationship between zeros and factors of polynomials◦ Use polynomial identities to solve problems◦ Solve systems of equations◦ Represent and solve equations and inequalities graphically◦ Analyze functions using different representations

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• Rational and Radical Relationships◦ Rewrite rational expressions

Create equations that describe numbers or relationships◦ Create equations that describe numbers or relationships ◦ Understand solving equations as a process of reasoning and explain the reasoning◦ Represent and solve equations and inequalities graphically◦ Interpret functions that arise in applications in terms of the context◦ Analyze functions using different representations

• Exponential and Logarithms◦ Write expressions in equivalent forms to solve problems◦ Analyze functions using different representationsAnalyze functions using different representations◦ Build new functions from existing functions◦ Construct and compare linear, quadratic, and exponential models and solve problems


• Trigonometric Functions◦ Analyze functions using different representations

Extend the domain of trigonometric functions using the unit circle◦ Extend the domain of trigonometric functions using the unit circle◦ Model periodic phenomena with trigonometric functions◦ Prove and apply trigonometric identities

• Mathematical Modeling◦ Create equations that describe numbers or relationships◦ Interpret functions that arise in applications in terms of the context◦ Analyze functions using different representations ◦ Build a function that models a relationship between two quantitiesBuild a function that models a relationship between two quantities◦ Build new functions from existing functions◦ Visualize relationships between two-dimensional and three-dimensional objects◦ Apply geometric concepts in modeling situations

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CCGPS Suggestions:1.Review the CCGPS. The teaching guide, curriculum map,

and standards can all be found in Learning Village on theand standards can all be found in Learning Village, on the Mathematics Program Page and on Georgia Standards.org

2.View the Fall 2011 Grade Level Webinar if you haven’t already seen it.already seen it.

3.Review this broadcast with your team to identify key areas of focus.

CCGPS Suggestions:4.Participate in the unit-by-unit webinars beginning in the

spring of 2013spring of 2013.

5.Structure time for grade level/content areas to use framework units as a guide for planning.

6.Plan to get together with your colleagues at the end of each CCGPS nit to anal e st dent ork samples andeach CCGPS unit to analyze student work samples and compare how student learning and performance look.

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Analytic Geometry Support:Now-

Fall 2011 Grade Level Webinar• Fall 2011 Grade Level Webinar• Standards Document• Teaching Guide• Curriculum MapComing soon-g• Framework Units (posting April 2012)• Unit-by-unit webinars

Takeaways

3 Things-3 gs

1. What’s new?

2. What’s different?

3 Wh t d t3. What resources and support are available for CCGPS mathematics?

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“The resources we need in order to grow as t h b d t ithi th itteachers are abundant within the community of colleagues. Good talk about good teaching is what we need…”

Parker PalmerCourage to Teach

Brooke Kline James Pratt

[email protected] [email protected]

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Thank you for participating in this CCGPS Professional Learning Session. We value your feedback. Please go to the following website, take the anonymous feedback survey, and complete the participation log to

receive a certificate of participation:

http://survey.sedl.org/efm/wsb.dll/s/1g10aIf you have questions, feel free to contact any of the English/Language Arts or

Mathematics staff at the following email addresses:Sandi Woodall, Georgia Mathematics Coordinator Kim Jeffcoat, Georgia ELA [email protected] [email protected]

James Pratt, Secondary Mathematics Susan Jacobs, Secondary ELA [email protected] [email protected]

Brooke Kline, Secondary Mathematics Sallie Mills, Elementary [email protected] [email protected]

Turtle Gunn Toms, Elementary Mathematics Andria Bunner, Elementary ELA

[email protected] [email protected]

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