Readings: Standards Deconstructed: High School Functions (located in Course Materials module) – pp. 16-37 Reading Response 6:? Respond to the fol

HS

COMMON CORE

MATHEMATICS

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HIGH SCHOOL FUNCTIONS

DECONSTRUCTED for CLASSROOM IMPACT

State Standards

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 3

MATHEMATICS

Introduction The Common Core Institute is pleased to offer this grade-level tool for educators who are teaching with the Common Core State Standards.

The Common Core Standards Deconstructed for Classroom Impact is designed for educators by educators as a two-pronged resource and tool 1) to help educators increase their depth of understanding of the Common Core Standards and 2) to enable teachers to plan College & Career Ready curriculum and classroom instruction that promotes inquiry and higher levels of cognitive demand.

What we have done is not all new. This work is a purposeful and thoughtful compilation of preexisting materials in the public domain, state department of education websites, and original work by the Center for College & Career Readiness. Among the works that have been compiled and/or referenced are the following: Common Core State Standards for Mathematics and the Appendix from the Common Core State Standards Initiative; Learning Progressions from The University of Arizona’s Institute for Mathematics and Education, chaired by Dr. William McCallum; the Arizona Academic Content Standards; the North Carolina Instructional Support Tools; and numerous math practitioners currently in the classroom.

We hope you will find the concentrated and consolidated resource of value in your own planning. We also hope you will use this resource to facilitate discussion with your colleagues and, perhaps, as a lever to help assess targeted professional learning opportunities.

Understanding the Organization

The Overview acts as a quick-reference table of contents as it shows you each of the domains and related clusters covered in this specific grade-level booklet. This can help serve as a reminder of what clusters are part of which domains and can reinforce the specific domains for each grade level.

Critical Areas of Focus is designed to help you begin to approach how to examine your curriculum, resources, and instructional practices. A review of the Critical Areas of Focus might enable you to target specific areas of professional learning to refresh, as needed.

For each domain is the domain itself and the associated clusters. Within each domain are sections for each of the associated clusters. The cluster-specific content can take you to a deeper level of understanding. Perhaps most importantly, we include here the Learning Progressions. The Learning Progressions provide context for the current domain and its related standards. For any grade except Kindergarten, you will see the domain- specific standards for the current

Math Fluency Standards K Add/subtract within 5

1 Add/subtract within 10

2 Add/subtract within 20 Add/subtract within 100 (pencil & paper)

3 Multiply/divide within 100 Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6 Multi-digit division Multi-digit decimal operations

7 Solve px + q = r, p(x + q) = r

8 Solve simple 2 x 2 systems by inspection

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4 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

grade in the center column. To the left are the domain-specific standards for the preceding grade and to the right are the domain-specific standards for the following grade. Combined with the Critical Areas of Focus, these Learning Progressions can assist you in focusing your planning.

For each cluster, we have included four key sections: Description, Big Idea, Academic Vocabulary, and Deconstructed Standard.

The cluster Description offers clarifying information, but also points to the Big Idea that can help you focus on that which is most important for this cluster within this domain. The Academic Vocabulary is derived from the cluster description and serves to remind you of potential challenges or barriers for your students.

Each standard specific to that cluster has been deconstructed. There Deconstructed Standard for each standard specific to that cluster and each Deconstructed Standard has its own subsections, which can provide you with additional guidance and insight as you plan. Note the deconstruction drills down to the sub-standards when appropriate. These subsections are:

� Standard Statement

� Standard Description

� Essential Question(s)

� Mathematical Practice(s)

� DOK Range Target for Learning and Assessment

� Learning Expectations

� Explanations and Examples

As noted, first are the Standard Statement and Standard Description, which are followed by the Essential Question(s) and the associated Mathematical Practices. The Essential Question(s) amplify the Big Idea, with the intent of taking you to a deeper level of understanding; they may also provide additional context for the Academic Vocabulary.

The DOK Range Target for Learning and Assessment remind you of the targeted level of cognitive demand. The Learning Expectations correlate to the DOK and express the student learning targets for student proficiency for KNOW, THINK, and DO, as appropriate. In some instances, there may be no learning targets for student proficiency for one or more of KNOW, THINK or DO. The learning targets are expressions of the deconstruction of the Standard as well as the alignment of the DOK with appropriate consideration of the Essential Questions.

The last subsection of the Deconstructed Standard includes Explanations and Examples. This subsection might be quite lengthy as it can include additional context for the standard itself as well as examples of what student work and student learning could look like. Explanations and Examples may offers ideas for instructional practice and lesson plans.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 5

MATHEMATICS

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PRACTICE EXPLANATION

MP.1 Make sense and persevere in problem solving.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

MP.2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Standards for Mathematical Practice in High School Mathematics Courses The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

6 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

PRACTICE EXPLANATION

MP.3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

MP.4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, 2-by-2 tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

MP.5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, computer algebra system, statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students interpret graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 7

MATHEMATICS

PRACTICE EXPLANATION

MP.6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

MP.7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

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

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)(x – 1) = 3.

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8 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

OVERVIEW

Functions describe situations where one quantity determines another. For example, the return on $10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the money is invested. Because we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models.

In school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression. For example, the time in hours it takes for a car to drive 100 miles is a function of the car’s speed in miles per hour, v; the rule T(v) = 100/v expresses this relationship algebraically and defines a function whose name is T.

The set of inputs to a function is called its domain. We often infer the domain to be all inputs for which the expression defining a function has a value, or for which the function makes sense in a given context.

A function can be described in various ways, such as by a graph (e.g., the trace of a seismograph); by a verbal rule, as in, “I’ll give you a state, you give me the capital city;” by an algebraic expression like f(x) = a + bx; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of the function models, and manipulating a mathematical expression for a function can throw light on the function’s properties.

Functions presented as expressions can model many important phenomena. Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with a constant term of zero describe proportional relationships.

A graphing utility or a computer algebra system can be used to experiment with properties of these functions and their graphs and to build computational models of functions, including recursively defined functions.

Connections to Expressions, Equations, Modeling, and Coordinates.

Determining an output value for a particular input involves evaluating an expression; finding inputs that yield a given output involves solving an equation. Questions about when two functions have the same value for the same input lead to equations, whose solutions can be visualized from the intersection of their graphs. Because functions describe relationships between quantities, they are frequently used in modeling. Sometimes functions are defined by a recursive process, which can be displayed effectively using a spreadsheet or other technology.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 9

MATHEMATICS OVERVIEW

Interpreting Functions (F-IF) � Understand the concept of a function and use function notation

� Interpret functions that arise in applications in terms of the context

� Analyze functions using different representations

Building Functions (F-BF) � Build a function that models a relationship between two quantities

� Build new functions from existing functions

Linear, Quadratic, and Exponential Models (F-LE) � Construct and compare linear, quadratic, and exponential models and solve problems

� Interpret expressions for functions in terms of the situation they model

Trigonometric Functions (F-TF) � Extend the domain of trigonometric functions using the unit circle

� Model periodic phenomena with trigonometric functions

� Prove and apply trigonometric identities

Mathematical Practices (MP) MP 1. Make sense of problems and persevere in solving them.

MP 2. Reason abstractly and quantitatively.

MP 3. Construct viable arguments and critique the reasoning of others.

MP 4. Model with mathematics.

MP 5. Use appropriate tools strategically.

MP 6. Attend to precision.

MP 7. Look for and make use of structure.

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

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The high school standards specify the mathematics that all students should study in order to be college and career ready. Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example: (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers).

All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students.

10 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

Domain HS Algebra I Mathematics I Geometry

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The Real Number System (RN) RN, 1, 2, 3

Quantities (Q) Q.1, 2, 3 Q.1, 2, 3

The Complex Number System (CN)

Vector Quantities and Matrices (VM)

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Seeing Structure in Expressions (SSEE) SSE.1a, 1b, 2, 3a, 3b, 3c SSE.1a, 1b

Arithmetic with Polynomials and Rational Expressions (APR) APR .1

Creating Equations (CED) CED. 1, 2, 3, 4 CED. 1, 2, 3, 4

Reasoning with Equations and Inequalities (REI)

REI. 1, 3, 4a, 4b, 5, 6, 7, 10, 11, 12 REI. 1, 3, 5, 6, 10, 11, 12

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Interpreting Functions (IF) IF, 1, 2, 3, 4, 5, 6, 7a, 7b, 7c, 8a, 8b, 9

IF. 1, 2, 3, 4, 5, 6, 7a, 7c, 9

Building Functions (BF) BF. 1a, 1b, 2, 3, 4a BF. 1a, 1b, 2, 3

Linear, Quadratic, and Exponential Models (LE) LE. 1a, 1b, 1c, 2, 3, 5 LE. 1a, 1b, 1c, 2, 3, 5

Trigonometric Functions (TF)

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Congruence (CO) CO. 1, 2, 3, 4, 5, 6, 7, 8, 12, 13 CO. 1-13

Similarity, Right Triangles, and Trigonometry (SRT) SRT. 1-11

Circle (C) C. 1-5

Expressing Geometric Properties with Equations (GPE) GEP. 4, 5, 7 GPE. 1, 2, 4-7

Geometric Measurement and Dimension (GMD) GMD. 1, 3, 4

Modeling with Geometry (MG) MG. 1, 2, 3

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Interpreting Categorical and Quantitative Data (D) ID. 1-3, 5-9 ID. 1-3, 5-9

Making Interference and Justifying Conclusions (IC)

Conditional Probability and the Rules of Probability (CP) CP. 1-9

Using Probability to Make Decisions (MD) MD. 6-7

High School Pathways for Traditional and Integrated Courses

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 11

MATHEMATICS

Mathematics II HS Algebra II Mathematics III 4th Courses (T) 4th Courses (I)

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RN, 1, 2, 3

CN.1, 2, 7, 8, 9 CN.1, 2, 7, 8, 9 CN. 8, 9 CN. 3, 4, 5, 6 CN. 3, 4, 5, 6

VM. 1, 2, 3, 4 (a-c), 5(a- b), 6, 7, 8, 9, 10, 11, 12

VM. 1, 2, 3, 4 (a-c), 5(a- b), 6, 7, 8, 9, 10, 11, 12

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SSE.1a, 1b, 2, 3a, 3b, 3c SSE. 1, 1b, 2, 4 SSE. 1, 1b, 2, 4

APR .1 APR .1-7 APR .1-7

CED. 1, 2, 4 CED. 1, 2, 3, 4 CED. 1, 2, 4

REI. 4a, 4b, 7 REI. 2, 11 REI. 2, 11 REI. 8, 9 REI. 8, 9

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IF. 4, 5, 6, 7a, 7b, 8a, 8b, 9 IF. 4, 5, 6, 7b, 7c, 7e, 8, 9 IF. 4, 5, 6, 7b, 7c, 7e, 8, 9 IF. 7d IF. 7d

BF. 1a, 1b, 3, 4a BF. 1b, 3, 4a BF. 1b, 3, 4a BF. 1c, 4b, 4c, 4d, 5 BF. 1c, 4b, 4c, 4d, 5

LE. 3 LE. 4 LE. 4

TF. 8 TF. 1, 2, 5, 8 TF. 1, 2, 5 TF. 3, 4, 6, 7, 9 TF. 3, 4, 6, 7, 9

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CO. 9, 10, 11

SRT. 1a, 1b, 2, 3, 4, 5, 6, 7, 8 SRT. 9, 10, 11

C. 1-5

GPE. 1, 2, 4 GPE. 3 GPE. 3

GMD. 1, 3 GMD. 4 GMD. 2 GMD. 2

MG. 1, 2, 3

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ID. 4 ID. 4

IC. 1-6 IC. 1-6

CP. 1-9

MD. 6-7 MD. 6, 7 MD. 6-7 MD. 1-5 MD. 1-5

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12 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

Functions Math 1 Math 2 Math 3 Math 4

Interpreting Functions (IF)

F.IF.1      

F.IF.2      

F.IF.3      

F.IF.4 F.IF.4 F.IF.4  

F.IF.5 F.IF.5 F.IF.5  

F.IF.6 F.IF.6 F.IF.6  

F.IF.7(a,e) F.IF.7(a,b) F.IF.7(b,c,e) F.IF.7(d)

 F.IF.8(a-b) F.IF.8  

F.IF.9  F.IF.9 F.IF.9  

Building Functions (IF) F.BF.1(a-b) F.BF.1(a-b) F.BF.1(b) F.BF.1 c

F.BF.2      

F.BF.3 F.BF.3 F.BF.3  

F.BF.4a F.BF.4a F.BF.4(b,c,d)

      F.BF.5

Linear, Quadratic, and Exponential Models (LE)

F.LE.1(a-c)      

F.LE.2      

F.LE.3  F.LE.3    

    F.LE.4  

F.LE.5      

Trigonometric Functions (TF)

   F.TF.8 F.TF.1  

    F.TF.2  

      F.TF.3

      F.TF.4

    F.TF.5  

      F.TF.6

      F.TF.7

      F.TF.9

Learning Progressions for Integrated Courses

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 13

MATHEMATICS

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Functions Algebra 1 (Grade 9) Geometry (Grade 10) Algebra 2 (Grade 11) PreCalculus (Grade 12)

Interpreting Functions (IF)

F.IF.1      

F.IF.2      

F.IF.3      

F.IF.4   F.IF.4  

F.IF.5   F.IF.5  

F.IF.6   F.IF.6  

F.IF.7(a,b,e)   F.IF.7(b,c,e) F.IF.7(d)

F.IF.8(a-b)   F.IF.8  

F.IF.9   F.IF.9  

Building Functions (IF)

F.BF.19a-b)   F.BF.1(b) F.BF.1 c

F.BF.2      

F.BF.3   F.BF.3  

F.BF.4(a)   F.BF.4a F.BF.4(b-c)

      F.BF.5

Linear, Quadratic, and Exponential Models (LE)

F.LE.1(a-c)      

F.LE.2      

F.LE.3      

    F.LE.4  

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